Answer:
the rise of the center of the rail will be given by =[tex]\sqrt{hypotenuse^{2}\hspace{0.15cm} -\hspace{0.15cm} base^{2} }[/tex]
= [tex]\sqrt{1500.065^{2} \hspace{0.15cm} -\hspace{0.15cm}1500^{2} }[/tex] m = 13.964 m
Step-by-step explanation:
i) the normal length of the railroad is 3 km = 3000 m.
ii) on a hot day day the railroad rail expands by 13cm = 0.13m
iii) the center of the railroad is at 1500 m which is half of the total length.
iv) the expansion of the railroad in either half of the road is 0.065m which is half of the total expansion.
v) each half of the railroad can be represented by a right angled triangle whose base will be = 1500 m and hypotenuse will be = 1500.065 m .
vi) the rise of the center of the rail will be given by =[tex]\sqrt{hypotenuse^{2}\hspace{0.15cm} -\hspace{0.15cm} base^{2} }[/tex]
= [tex]\sqrt{1500.065^{2} \hspace{0.15cm} -\hspace{0.15cm}1500^{2} }[/tex] m = 13.964 m
The center of the rail would rise about 16.06 meters above the ground.
To find out how much the center of the rail rises above the ground due to expansion:
we can consider that the rail expands uniformly along its length.
If it expands by 13 cm over a length of 3 km, we can calculate the rise at the center using trigonometry, considering that the expansion causes the rail to form a very shallow arc.
The expansion forms a segment of a circle, where the rail length is the arc length, and the rise at the center is the height of the segment. We can use the formula for the height of a segment of a circle:
[tex]\[ h = r - \sqrt{r^2 - \left(\frac{l}{2}\right)^2} \][/tex]
Where:
h is the height of the segment (the rise at the center),
r is the radius of the circle (which we can approximate as the radius of the Earth, as the rail is very long compared to its curvature),
l is the arc length of the segment (the expansion of the rail).
Given that the Earth's radius is approximately 6371 km, we need to convert all lengths to meters:
[tex]\[ r = 6371 \times 1000 \text{ meters} \][/tex]
[tex]\[ l = 3 \times 1000 \times 100 \text{ meters} \][/tex]
[tex]\[ h = 6371 \times 1000 - \sqrt{(6371 \times 1000)^2 - \left(\frac{3 \times 1000 \times 100}{2}\right)^2} \][/tex]
[tex]\[ h \approx 6371000 - \sqrt{(6371000)^2 - (150000)^2} \][/tex]
[tex]\[ h \approx 6371000 - \sqrt{40641464900000 - 22500000000} \][/tex]
[tex]\[ h \approx 6371000 - \sqrt{40641442400000} \][/tex]
[tex]\[ h \approx 6371000 - 6370983.94 \][/tex]
[tex]\[ h \approx 16.06 \text{ meters} \][/tex]
The points obtained by students of a class in a test are normally distributed with a mean of 60 points and a standard deviation of 5 points.
About what percent of students have scored more than 75 points?
0.15
2.5
15.5
34
Answer:
Option A) is correct.
Step-by-step explanation:
We need to use normalcdf command to find the probability that the variable would fall into a certain interval that we would supply.
As
The points obtained by students of a class in a test are normally distributed with a mean of 60 pointsand
a standard deviation of 5 pointsAnd we have to determine the percent of students have scored more than 75 points.
So,
Mean = μ = 60
Standard Deviation = σ = 5
As we have to determine the percent of students have scored more than 75 points.
Hence,
normalcdf(75,100,60,5) = .0013
Converting on percentage → .0013 × 100 = 0.1
Therefore, option A) is correct as 0.1 percent of students have scored more than 75 points.
Keywords: distribution, mean, median
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Producing a musical costs $88,000 plus $5900 per performance. One sold-out performance earns $7500 in revenue. If every performance sells out, how many performances are needed to break even?
Answer: 55 performances are needed to break even.
Step-by-step explanation:
Let x represent the number of musical performances that are made.
Producing a musical costs $88,000 plus $5900 per performance. This means that the total cost of x musical performances would be
88000 + 5900x
One sold-out performance earns $7500 in revenue. If every performance sells out, it means that the total revenue would be
7500 × x = 7500x
In order to break even, total cost would be equal to total revenue. Therefore,
7500x = 88000 + 5900x
7500x - 5900x = 88000
1600x = 88000
x = 88000/1600 = 55
Which of the following combinations of fishing lures and duck decoys is unobtainable to Big Lake Bob in one week's worth of carving? 80 fishing lures and 5 duck decoys 40 fishing lures and 20 duck decoys 50 fishing lures and 10 duck decoys 20 fishing lures and 24 duck decoys c. Which of the following combinations of fishing lures and duck decoys is an efficient combination? 20 fishing lures and 24 duck decoys 50 fishing lures and 0 duck decoys 10 fishing lures and 40 duck decoys 16 fishing lures and 30 duck decoys?
Answer:
b) 40 fishing lures and 20 duck decoys
c) 20 fishing lures and 24 duck decoys
Question:
Attached is the graph to be used to answer the questions.
Step-by-step explanation:
b) Any point above the straight line graph PPF is unobtainable, while points below or on the line graph are obtainable.
Finding each of the points on the graph and determine whether it is above, on or below the line graph.
- Only one point is above the graph (40 fishing lures and 20 duck decoys )
Therefore, 40 fishing lures and 20 duck decoys is unobtainable.
c) for an efficient combination the points must be on the line of the graph (PPF).
The only point that falls on the line of the graph is (20 fishing lures and 24 duck decoys)
Therefore, 20 fishing lures and 24 duck decoys is an efficient combination.
An airline charges the following baggage fees: $25 for the first bag and $35 for the second. Suppose 54% of passengers have no checked luggage, 34% have one piece of checked luggage and 12% have two pieces. We suppose a negligible portion of people check more than two bags.
(a) Build a probability model, compute the average revenue per passenger, and compute the
corresponding standard deviation.
(b) About how much revenue should the airline expect for a flight of 120 passengers? With what
standard deviation? Note any assumptions you make and if you think they are justified.
Answer:
The average revenue per passenger = E(x) = $15.70
[tex]\sigma = \sqrt{\sigma^{2}} = \sqrt{398} \approx 19.95[/tex]
b)$1,884 ± $20
Step-by-step explanation:
a)
Refer attached fig for probability model,
i)The average revenue per passenger = E(x) = $15.70
[tex]\sigma^{2} = V[X] \sum _{x} (x-\mu )^{2} \times P(X=x)[/tex]
[tex]\sigma^{2} = (0-15.7)^{2} \times 0.54+(25-15.7)^{2} \times 0.34 +(60-15.7)^{2} \times 0.12[/tex]
[tex]\sigma^{2} = 133.1 + 29.4 + 235.5 = 398[/tex]
Standard Deviation = [tex]\sigma = \sqrt{\sigma^{2}} = \sqrt{398} \approx 19.95[/tex]
ii) average revenue per passenger = E(x) = $15.70
b) Revenue should the airline expect for a flight of 120 passengers
revenue = 120 * $15.70 = $1,884 ± $20 (or 19.95 rounded up)
use same σ
The average revenue per passenger is $15.2 and the standard deviation is $18.45. For a flight of 120 passengers, the airline may expect around $1824 with standard deviation of $202.33. We're assuming that each passenger's baggage habits are independent.
Explanation:To solve this, first we'll construct a probability model. Let X be a random variable denoting the revenue from each passenger. Then X can take three possible outcomes: $0 (with probability 0.54), $25 (with probability 0.34) or $60 (with probability 0.12).
The average revenue from each passenger E(X), also known as the expected value of X, is calculated as follows:
E(X) = 0*0.54 + 25*0.34 + 60*0.12 = $15.2
The standard deviation of X, denoted σ(X), is computed as follows:
σ(X) = sqrt[(0-15.2)^2*0.54 + (25-15.2)^2*0.34 + (60-15.2)^2*.12] = $18.45
Given this, for a flight of 120 people, the total expected revenue is 120*$15.2 = $1824 and the standard deviation of total revenue would be sqrt(120)*$18.45 = $202.33.
The underlying assumption here is that the baggage habits of passengers are independent, which is reasonable in this case.
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Solve.
{m+n=5m−n=3
Use the substitution method.
The values of m and n are [tex]m=4[/tex] and [tex]n=1[/tex]
Explanation:
The two equations are [tex]m+n=5[/tex] and [tex]m-n=3[/tex]
From the equation [tex]m-n=3[/tex], we shall find the value of m such that
[tex]m=n+3[/tex]
Substituting the value of m in [tex]m+n=5[/tex], we get,
[tex]\begin{array}{r}{n+3+n=5} \\{2 n+3=5}\end{array}[/tex]
Subtracting both sides by 3,
[tex]2n=2[/tex]
Dividing both sides by 2, we get,
[tex]n=1[/tex]
Substituting [tex]n=1[/tex] in [tex]m-n=3[/tex]
[tex]\begin{array}{r}{m-1=3} \\{m=4}\end{array}[/tex]
Thus, the values of m and n are [tex]m=4[/tex] and [tex]n=1[/tex]
A researcher wishes to estimate the average amount spent per person by visitors to a theme park. He takes a random sample of forty visitors and obtains an average of $28 per person.
a.What is the population of interest?
b.What is the parameter of interest?
c.Based on this sample, do we know the average amount spent per person byvisitors to the park? Explain fully
Answer: a. Number of visitors
b. average([tex]\mu[/tex]) amount spent per person by visitors to the park
c. Yes , we know the average amount spent per person byvisitors to the park. It is $28 per person.
Step-by-step explanation:
A population is a group of all members according to the researchers's subject of interest.A parameter is a number that gives the measure of a factor generated from population (for ex Population mean ([tex]\mu[/tex]) , population proportion (p) ).A sample is a finite subset of the population that represents the population in an analysis.A statistic is a number that gives the measure of a factor generated from sample (for ex sample mean ([tex]\overline{x}[/tex]) ,sample proportion ([tex]\hat{p}[/tex]) ).For the given situation, A researcher wishes to estimate the average amount spent per person by visitors to a theme park.
Population by researcher's point of interest : "Number of visitors"
Parameter of interest : "average([tex]\mu[/tex]) amount spent per person by visitors to the park"
Since he takes a random sample of forty visitors and obtains an average of $28 per person.
Here , sample : forty visitors
And average amount spent per person by visitors to the park from sample :
[tex]\overline{x}= [/tex] $28 per person.
The sample means is the best point-estimate for the population mean.
So , the best point-estimate for average amount spent per person by visitors to the park here is $28 per person.
The population of interest is:
Number of visitors
The parameter of interest is:
Average amount spent on each person by visitors in the parkBased on the sample, the way know the average amount spent per person by visitors to the park is:
$28 per personPopulation of InterestThis refers to the select sample which a researcher tries to draw conclusions from.
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Suppose your group makes several more measurements of the same quantity and then calculates the standard error on the mean again using all of your measurements. How would you expect the standard error to change?
Answer:
I would expect the standard error to decrease, because sample size increases.
Step-by-step explanation:
Standard error on the mean can be calculated using the equation:
[tex]\frac{s}{\sqrt{N}}[/tex] where
s is the population standard deviationN is the sample sizeTherefore, if the group makes several more measurements then the sample size increases, which makes the standard error on the mean smaller.
The standard error is expected to decrease when more measurements are made and used to recalculate it.
Explanation:The standard error of the mean is a measure of the variability of sample means. As you obtain more measurements and calculate the standard error again using all the measurements, you would expect the standard error to decrease. This is because the more measurements you have, the more precise your estimate of the true mean becomes.
When calculating the standard error, you divide the standard deviation by the square root of the sample size. The standard deviation measures the variability of the individual measurements, while the square root of the sample size represents the precision of the mean. As the sample size increases, the precision increases, resulting in a smaller standard error.
For example, let's say you initially had a small sample size of 10 and calculated a standard error of 2. If you then collect more measurements and increase the sample size to 20, you would expect the standard error to decrease, potentially to a value smaller than 2.
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Use f(x) = 1 2 x and f -1(x) = 2x to solve the problems.
f−1(−2) =
f(−4) =
f(f−1(−2)) =
Answer:
The answer to your question is below
Step-by-step explanation:
Functions
f(x) = 12x f⁻¹(x) = 2x
a) f⁻¹(-2) = 2(-2)
f⁻¹(-2) = -4
b) f(-4) = 12x
f(-4) = 12(-4)
f(-4) = -48
c) f(f⁻¹(-2)) =
f(f⁻¹(x)) = 12(2x) = 24x
f(f⁻¹(-2)) = 24(-2) = -48
I think your functions are wrong they must be f(x) = 1/2x f⁻¹(x) = 2x
a) f⁻¹(-2) = 2(-2)
= -4
b) f(-4) = 1/2(-4)
= -2
c) f(f⁻¹(x)) = 1/2(2x)
= x
f(f⁻¹(-2)) = -2
Answer:
1)= -4
2)= -2
3)= -2
Step-by-step explanation:
A right triangle has sides of length 3 feet, 4 feet, and 5 feet what is its perimeter? What is its area?
Answer:
Perimeter= 12 feet Area= 6 feet (squared)
Step-by-step explanation:
Perimeter= Adding all the sides together, that means it is 3+4+5= 12 feet
Area= Hight times base divided by 2. Because hight times base is area of a rectangle, and a right triangle (and all other triangles) is half of it. Because it is a right triangle, its hight must be 3 or 4, and base also 3 or 4. But no the same number twice, so 3*4=12, 12/2=6 feet (squared).
I hope that helped! =)
If A is independent of B and B is independent of C, then A is independent of C. Prove this statement or give a counterexample if it is false.
Answer:
This is true.
Step-by-step explanation:
This would be known as the transitive property.
If A = B, and B = C, then A must equal C
Let me explain.
If A = 2, and B = 2, and A = C, C must also equal 2.
If something is equal then it's the same, so it makes sense for A = C if B is the same value as A and C.
A good way to recognize the property is if you see a number/letter appear twice, and they both equal different things. Then you see the two different letters are equal to each other. Hopefully this helped some.
The principle of independence is not transitive. If A and B are independent, and B and C are independent, it does not necessarily mean that A and C are independent. This can be proven through a counterexample involving coin tosses.
Explanation:The statement 'If A is independent of B and B is independent of C, then A is independent of C' is not necessarily true. This is so because independence is not transitive.
Consider the following counterexample:
A fair coin is tossed to decide two independent events A and B. Let A be the event of getting a 'head' and B be the event of getting a 'tail'.Now, let B and C also be two independent events. Assuming C to be the event of getting a 'head' on the second toss.In this example, event A (getting head in first toss) is independent of B (getting tail in first toss), and B is independent of C (getting head in second toss). However, A is clearly not independent of C because both are 'getting head' on their respective tosses.
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A town has a population of 1600 people at time ????=0. In each of the following cases, write a formula for the population P, of the town as a function of year ????.
Answer:
a) [tex] P(t) =1600 +50t[/tex]
b) [tex] P(t) = 1600 (1.05)^t[/tex]
Step-by-step explanation:
Assuming the complete question : "A town has a population of 1000 people at time t = 0. In each of the following cases, write a formula for the population, P, of the town as a function of year t. (a) The population increases by 50 people a year. (b) The population increases by 5% a year."
Part a
For this case we can use a linear model in order to estimate the population size since we have a fixed increase each year. So our model would be given by:
[tex] P(t) = P_o + b t[/tex]
Where [tex] b =50[/tex] on this case since represent the increase per year of the slope for the linear model. And the initial amount is [tex] P_o = 1600[/tex], so then the model is:
[tex] P(t) =1600 +50t[/tex]
Part b
For this case we have a rate of increase and when we have this the lineal model is not the most appropiate. So then we can use an exponential model given by:
[tex] P(t) =P_o b^{t}[/tex]
Where [tex] P_o = 1600[/tex] represent the initial population and for this case b is the rate of increase [tex] b = 1+0.05 = 1.05[/tex] since each year we have an increase of 5% and t is the time. So then the model is given by:
[tex] P(t) = 1600 (1.05)^t[/tex]
The population of a town can be calculated through formulas for linear growth (P = P0 + rt) and exponential growth (P = P0 * e^(rt)), depending on whether the population grows by a fixed amount or percentage each year. To use these formulas, the specific growth rate of the town is needed.
Explanation:The subject of the question is about writing formulas for population growth. Let's denote the initial population as P0, the rate of growth as r and the time in years as t. Here are two common formulas:
Linear growth: P = P0 + rt. If the population grows by a fixed amount each year, you can use this formula where r is the number of people the population grows by each year.Exponential growth: P = P0 * e^(rt). If the population grows by a fixed percentage each year, you can use this formula where r is the growth rate expressed as a decimal.To use these formulas, you'd need to know the specific growth rate of the town. It's also important to remember that actual population changes can be more complex and may not follow these exact formulas.
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At a real estate agency, an agent sold a house for $392 comma 000. The commission rate is 7.5% for the real estate agency and the commission rate for the agent is 20% of the amount the real estate agency gets. How much did the agency make on the house? How much did the agent earn in commission? The agency made $ nothing on the house.
Answer:
Step-by-step explanation:
At a real estate agency, an agent sold a house for $392000.
The commission rate is 7.5% for the real estate agency. This means that the amount of money that the real estate agency earn would be
7.5/100 = 392000 = 0.075 × 392000 = $29400
the commission rate for the agent is 20% of the amount the real estate agency gets. This means that the amount that the agent would earn is 20/100 × 29400 = 0.2 × 29400 = $5880
Ajar contains 8 blue marbles, 6 red marbles, and 10 green marbles. You pick one marble
from the jar. Find the theoretical probability, P(blue or green). Write your answers as a
decimal rounded to nearest hundredth place.
Answer:
.75 is your answer
Step-by-step explanation:
The math club makes 35 bars of laundry soap a week and sells these at $20 each before the soap could be sold , the pupils found that 6 bars were destroyed by mice. How much will the total sale at the end of a four week month?
Answer:
Step-by-step explanation:
The math club makes 35 bars of laundry soap a week and sells these at $20 each. This means that the total number of bars of laundry soap made in a 4 week month would be
4 × 35 = 140 bars
If the pupils found that 6 bars were destroyed by mice, the total number of bars of soap left would be
140 - 6 = 134 bars
Therefore, the total sale at the end of a four week month would be
134 × 20 = $2680
The total sales at the end of a 4-week month, given that 6 out of 35 soap bars were destroyed by mice every week and the price of each soap bar is $20, would be $2320.
Explanation:The math club initially makes 35 bars of soap every week. However, due to the unforeseen mouse problem, 6 bars are rendered unsalable each week. Therefore, the club is effectively selling only 29 bars per week. Because the selling price of each bar is $20, each week's sales amount to 29 bars x $20/bar = $580.
Over 4 weeks, the total sales would then be $580/week x 4 weeks = $2320. So, the total amount of sales at the end of a 4-week month, after accounting for the mice damage, would be $2320.
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Solve the equation, if possible. (If all real numbers are solutions, enter REALS. If there is no solution, enter NO SOLUTION.) x2 = (x + 3)(x − 3) + 9
Answer:
REALS. all real numbers are solution for the equation
Step-by-step explanation:
for the equation
x² = (x + 3)*(x − 3) + 9
distributing the terms in the parenthesis
x² = (x + 3)*(x − 3) + 9 = [x² - 3*x + 3*x - 3*3 ] + 9
x² = x² - 9 + 9
0 = 0
since this statement will be true regardless of the value of x , then the equation has solution for all real numbers .
A nationwide survey of 1000 adults found that 50% of respondents favored a plan to break up the 12 megabanks, which then controlled about 69% of the banking industry. Complete parts (a) and (b) below.
a. Identify the population and sample for this study.
b. Is the percentage provided a descriptive statistic or an inferential statistic? Explain your answer.
a. The population for the study is a total of 1000 adults, so the 50% that favored the plan made up a total of 500 people, since 500 is 1/2 (or 50%) of 1000.
b. The percentage provided is an inferential statistic. Inferential statistics are made by taking a sample of a population and using that sample as data used to make inferences/predictions. Since the percentage provided is a sample of a population being used to make a prediction, it is an inferential statistic.
The population is all adults in the nation, and the sample is the 1000 adults surveyed. The provided percentage is a descriptive statistic, as it summarizes data from the sample.
Explanation:In the context of this problem, the population refers to all adults across the nation, while the sample refers to the 1000 adults who were surveyed.
The percentage mentioned in the survey (50% in favor of breaking up the 12 megabanks) is a descriptive statistic. A descriptive statistic summarizes and represents data from a sample, which in this case is the surveyed group of 1000 adults. It does not aim to infer or predict facts about the larger population.
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A deck of cards contains 52 cards, of which 4 are aces. You are offered the following. Draw one card at random from the deck. You win $11 if the card drawn is an ace, otherwise you lose $1. If you make this wager very many times, what will the mean outcome be_____________.
Answer:
[tex] E(X) = 11 *\frac{4}{52} - 1*\frac{48}{52} = -\frac{1}{13}[/tex]
So we expect to lose approximately [tex] 1/13[/tex] for this game.
Step-by-step explanation:
For this case we can find the probability of win like this:
[tex] p_w = \frac{4}{52}[/tex]
Since we have 4 aces each time in a total of 52.
And the probability of loss is given by:
[tex] p_l = \frac{48}{52}[/tex]
In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".
And we can find the expected value with this formula:
[tex] E(X) = \sum_{i=1}^n X_i P(X_i) [/tex]
Where X on this case represent the random variable "Amount of money win or loss in the game", for this case we can replace and we got:
[tex] E(X) = 11 *\frac{4}{52} - 1*\frac{48}{52} = -\frac{1}{13}[/tex]
So we expect to lose approximately [tex] 1/13[/tex] for this game.
The mean outcome of this wager is -$0.62, indicating an expected average loss of $0.62 per game.
Explanation:To determine the mean outcome of this wager, we need to calculate the expected value. The probability of drawing an ace from the deck is 4/52, since there are 4 aces out of 52 cards. The payoff for drawing an ace is $11, while the payoff for not drawing an ace is -$1. The expected value is calculated by multiplying the probability of each outcome by its corresponding payoff and summing them up:
Expected value = (Probability of outcome 1 x Payoff for outcome 1) + (Probability of outcome 2 x Payoff for outcome 2)
Expected value = (4/52 x $11) + (48/52 x -$1)
Expected value = -$0.62
Therefore, if you play this game repeatedly, you would expect to lose $0.62 per game, on average. The expected value indicates an expected average loss, so it is not advisable to play this game to win money.
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A consumer has $100 per day to spend on product A, which has a unit price of $7, and product B, which has a unit price of $15. What is the slope of the budget line if good A is on the horizontal axis and good B is on the vertical axis?
The slope of the budget line for a consumer who has $100 to spend per day, with product A at $7/unit on the horizontal axis and product B at $15/unit on the vertical axis, is -0.47. More consumption of one good requires a sacrifice in the consumption of the other.
Explanation:The subject of this question is the slope of the budget line in microeconomics, which represents the trade-off between two goods, product A and product B. It can be determined by dividing the price of the good on the horizontal axis (product A) by the price of the good on the vertical axis (product B). Hence, the slope of the budget line is $7/$15 = 0.47 (rounded to two decimal places). The slope is negative reflecting the fact that to obtain more of one good, a consumer needs to give up some of the other.
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The slope of the budget line is -7/15 or approximately -0.467.
To determine the slope of the budget line for a consumer with $100 per day to spend on product A and product B, we need to use the given prices and quantities. The unit price of product A is $7, and the unit price of product B is $15.We place the quantity of product A on the horizontal axis and the quantity of product B on the vertical axis. The slope of the budget line is given by the formula:The slope of the budget line = - (price of the good on the horizontal axis) / (price of the good on the vertical axis).In this case, the slope = - (7 / 15).
After calculating, we get the slope as - 7/15 or approximately -0.467. This slope indicates the rate at which the consumer can trade product A for product B while staying within their budget.
What is the mathematical meaning of each symbol below? Give an example of each using numbers and/or algebraic expressions. ∼ ∪ ∩ ∅ ≠ !
Answer:
∼ : This symbol denotes similarity
∪ : This symbol means union
∩ : This symbol means Intersection
∅ : means an empty set
≠ : this symbol mean not equal to
! : This symbol means factorial
Step-by-step explanation:
Given symbols in the question:
∼ : This symbol denotes similarity
i.e the resemblance or likeness.
∪ : This symbol means union
i.e
A∪B element that belong to set A or set B
∩ : This symbol means Intersection
i.e
A∩B = Element that belong to set A and set B
∅ : means an empty set
For example
if A = { 1, 2, 3 } B = { 4, 5, 6 }
Then,
A∩B = ∅
≠ : this symbol mean not equal to
i.e
LHS is not equal to RHS
! : This symbol means factorial
i.e
If we write 3!
we solve it as 3! = 3 × 2 × 1
Lilah is moving from Portland to Seattle. It takes her 3 hours to go by train. Mason leaves the train station in Portland and drives to the train station in Seattle with all Lilah's boxes in his car. It takes him 2.4 hours to get to Seattle, driving at 15 miles per hour faster than the speed of the train. Find Lilah's speed on the train and Mason's speed.
Answer:Lilah's speed on the train is 60 mph and Mason's speed is 75 mph
Step-by-step explanation:
Let x represent the speed of Lilah.
Lilah is moving from Portland to Seattle. It takes her 3 hours to go by train.
Distance = speed × time
Distance covered by Lilah in moving from Portland to Seattle would be
3 × x = 3x
It takes him 2.4 hours to get to Seattle, driving at 15 miles per hour faster than the speed of the train.
It means that Mason's speed is
x + 15
Distance covered by Mason would be
2.4(x + 15) = 2.4x + 36
Since the distance from Portland to Seattle is the same, then
3x = 2.4x + 36
3x - 2.4x = 36
0.6x = 36 x = 36/0.6 = 60 mph
Mason's speed would be 60 + 15 = 75 mph
kristen invests money in a bank and makes no additional deposits or withdrawals. The bank pays 1.4% interest compounded annually. To the nearest tenth of a year, how long must she leave the money in the bank for it to double? SOLVE ALGEBRAICALLY.
Answer:approximately 50 years.
Step-by-step explanation:
Let $P represent the initial amount that she deposited. It means that principal,
P = $P
It was compounded annually. This means that it was compounded once in a year. So
n = 1
The rate at which the principal was compounded is 1.4%. So
r = 1.4/100 = 0.014
The formula for compound interest is
A = P(1+r/n)^nt
A = total amount in the account at the end of t years. For the initial amount to double, it means that
A = 2P
Therefore
2P = P (1+0.014/1)^1×t
2P/P = (1.014)^t
2 = (1.014)^t
Taking log to base 10 of both sides, it becomes
Log 2 = log 1.014^t
Log 2 = tlog 1.014
0.301 = 0.006t
t = 0.301/0.006 = 50.2 years
Answer: 51.4 years
Step-by-step explanation:
There is what we call rule of 72 in Accounting to find the time a sum of money will double if it is compounded annually at any given rate . The formula is given as :
t = [tex]\frac{72}{r}[/tex]
where t is the time and r is the rate.
t = ?
r = 1.4
Therefore :
t = 72/1.4
t = 51.4286
to the nearest tenth
t = 51. 4 years
A wildlife biologist determines that there are approximately 200 deer in a region of a national park. The population grows at a rate of 7% per year. What is an exponential function that models the expected population?
Answer:
The exponential function is Y = 200 [1.07]ˣ
Step-by-step explanation:
Let Y represent the expected population
Let P represent the current population
Let r represent the population growth rate
let x represent the number of years or nth year.
The compound interest expression can used to derive the exponential function can be represented as follows
Y = P [1 + (r ÷ 100)]ˣ
Y = 200 [ 1 + (7 ÷ 100)]ˣ
Y = 200 [ 1 + (0.07)]ˣ
Y = 200 [1.07]ˣ
Final answer:
To model the population growth of deer in a national park, an exponential function P(t) = 200 * (1 + 0.07)^t is used. This reflects a starting number of 200 deer, with growth at a rate of 7% per year, allowing for future population predictions.
Explanation:
A wildlife biologist determines that there are approximately 200 deer in a region of a national park. The population grows at a rate of 7% per year. To model the expected population using an exponential function, we use the general formula for exponential growth P(t) = P0 * (1 + r)^t, where:
P(t) represents the population at time t years,P0 is the initial population size,r is the annual growth rate as a decimal,t is the number of years.Given that the initial population (P0) is 200 deer and the annual growth rate (r) is 7% or 0.07, the exponential growth model to predict future population sizes can be written as P(t) = 200 * (1 + 0.07)^t.
This function can be used to calculate the expected number of deer in this region of the national park for any number of years into the future, allowing biologists and park management to make informed decisions regarding wildlife conservation and management.
Use vectors to show that, for any triangle1 in R², the points2 that are 2/3 of the way from a vertex to the midpoint of the opposite side are all the same point. This point is called the centroid of the triangle.
True / False.
Final answer:
The statement is true; the centroid of a triangle is the point that lies 2/3 of the way from each vertex to the midpoint of the opposite side. The centroid is the same point for all three vertices, which can be shown using vector operations that equate to (A+B+C)/3 for a triangle with vertices A, B, and C.
Explanation:
The statement that points lying 2/3 of the way from a vertex to the midpoint of the opposite side for any triangle in ℝ² (the real number plane) are all the same point, and this point is called the centroid of the triangle, is True. To prove this, consider a triangle ABC with vertices A, B, and C. If M is the midpoint of side BC, N is the midpoint of side AC, and O is the midpoint of side AB, then the points two-thirds of the way from the vertices A, B, and C to the midpoints of the opposite sides, respectively, can be represented by the vectors (2M+A)/3, (2N+B)/3, and (2O+C)/3.
Applying the midpoint formula and vector addition, we find that these points are equivalent to (B+C)/3, (A+C)/3, and (A+B)/3 respectively, which, when simplified, all yield to (A+B+C)/3, illustrating that the points coincide and thus prove the existence of a single centroid.
In a geometric context, the centroid is the point where the triangle’s medians intersect, and it also serves as the triangle's center of mass. The centroid divides each median in a 2:1 ratio, with the larger portion being closer to the vertex. This property is intrinsic to triangles and can be demonstrated using vector algebra or geometric arguments.
An annuity promises that, if the annuitant dies before receiving payments equal to the correct value, the payments will be continued to a beneficiary until an amount equal to the contract value has been paid. This type of annuity is called:_______
The type of annuity in the question is known as a refunded annuity. It ensures that the entire contract value is utilized in case the annuitant dies prematurely. The payments would continue to a beneficiary until the total contract value has been paid.
Explanation:The type of annuity described in the question, where payments continue to a beneficiary until the contract value is paid out if the annuitant dies prematurely, is referred to as a refunded annuity. Annuities are often used in retirement planning. This specific type of annuity gives protection to the annuitant's investment, ensuring that the whole amount of the contract value is utilized even if the annuitant dies before receiving the full value in payments. For instance, if an annuitant bought an annuity worth $200,000 and dies after receiving only $50,000, the beneficiary would continue to receive payments until the remaining $150,000 is paid out.
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A cash refund annuity is an annuity that promises to continue payments to a beneficiary until the contract value has been paid if the annuitant dies before that.
Explanation:The type of annuity described in the question, where the payments continue to a beneficiary until an amount equal to the contract value has been paid if the annuitant dies before that, is called a cash refund annuity.
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Meal tickets at the circus cost $4.00 for children and $12.00 for adults. If 1,650 meal tickets were bought for a total of $14,200, how many children and how many adults bought meal tickets? Just type the number of children followed by the number of adults with a comma in between.
Answer:
700 children tickets, 950 adult tickets
Step-by-step explanation:
Let x and y be the number of tickets sold for children and adults, respectively. We know that the total number of tickets sold is 1650. So x + y = 1650, or x = 1650 - y
Also the total ticket revenue is $14200, at $12 per adult and $4 per child. We have:
4x + 12y = 14200
We can substitute x = 1650 - y
4(1650 - y) + 12y = 14200
6600 - 4y + 12y = 14200
6600 + 8y = 14200
8y = 14200 - 6600 = 7600
y = 7600 / 8 = 950 adult tickets
x = 1650 - 950 = 700 children tickets
The area of an equilateral triangle is given by A =3^1/2/4s^2. Find the length of the side s of an equilateral triangle with an area of 12^1/2 square inches.
Answer:
B on edu2020
Step-by-step explanation:
What is the solution of the equation (x – 5)2 + 3(x – 5) + 9 = 0? Use u substitution and the quadratic formula to solve.
x = StartFraction negative 3 plus-or-minus 3 i StartRoot 3 EndRoot Over 2 EndFraction
x = StartFraction 7 plus-or-minus 3 i StartRoot 3 EndRoot Over 2 EndFraction
x = 2
x = 8
Answer:
Option 2 - [tex]x=\frac{7\pm3\sqrt{3}i}{2}[/tex]
Step-by-step explanation:
Given : Equation [tex](x-5)^2+3(x-5)+9=0[/tex]
To find : What is the solution of the equation ?
Solution :
Using substitution method,
Let y=x-5
[tex]y^2+3y+9=0[/tex]
Using quadratic formula, [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Here, a=1, b=3 and c=9
[tex]y=\frac{-3\pm\sqrt{3^2-4(1)(9)}}{2(1)}[/tex]
[tex]y=\frac{-3\pm\sqrt{-27}}{2}[/tex]
[tex]y=\frac{-3\pm3\sqrt{3}i}{2}[/tex]
Substitute back,
[tex]x=\frac{-3\pm3\sqrt{3}i}{2}+5[/tex]
[tex]x=\frac{-3\pm3\sqrt{3}i+10}{2}[/tex]
[tex]x=\frac{7\pm3\sqrt{3}i}{2}[/tex]
Therefore, option 2 is correct.
Answer:
Option 2
Step-by-step explanation:
I just took the test and got it right.
The population of Asia is approximately 4.5 x 10^9. If the population of Earth is about double that of Asia, what is the approximate population of Earth?
Answer:
9x10^9
Step-by-step explanation:
2x4.5x10^9
just multiply by two.
Answer:
2,000,000,009
Step-by-step explanation:
2(4.5 x 10^9)
Distribute the 2
(2 x 4.5) +2(10^9)
= 9 + 2,000,000,000
Find a such that the line x = a divides the region bounded by the graphs of the equations into two regions of equal area. (Round your answer to three decimal places.)
y = x, y = 8, x = 0
a = _____.
Answer:
a≈2.343
Step-by-step explanation:
View Image
Integrating an equation from boundary x₀ to x₁ gives you the area underneath that boundary.
So to find the boundary that split the equation into 2 equal areas, the boundary must lie somewhere between the 2 place to want to split up. In other word, a is the end boundaries of the first integral and the starting boundary of the second integral.
Since the two area must equal to each other, set the two integral equal to each other and solve for a.
The value of a for which x=a will divide the enclosed area into two equal areas will be 2.43.
Let us say x=a divides the region into two equal areas.
From the attached diagram,
Point D≡(a,8)
Point E≡(a, a)
The length of DE will be 8-a.
As we know that triangles ADE and ABO will be similar
So, [tex]\frac{areaADE}{areaABO} = (\frac{DE}{BO}) ^{2}[/tex]
[tex]\frac{1}{2} = (\frac{a-8}{8}) ^{2}[/tex]
[tex]a^2-16a+32 = 0[/tex]
a =13.66(Not Possible)
a= 2.34
Therefore, The value of a for which x=a will divide the enclosed area into two equal areas will be 2.43.
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What is the sum of all of the perfect squares between
15
and
25
, inclusive, minus the sum of all of the other numbers between
15
and
25
,
inclusive?
Answer:
(-138) is the answer.
Step-by-step explanation:
Perfect square numbers between 15 and 25 inclusive are 16 and 25.
Sum of perfect square numbers 16 and 25 = 16 + 25 = 41
Sum of the remaining numbers between 15 and 25 inclusive means sum of the numbers from 17 to 24 plus 15.
Since sum of an arithmetic progression is defined by the expression
[tex]S_{n}=\frac{n}{2}[2a+(n-1)d][/tex]
Where n = number of terms
a = first term of the sequence
d = common difference
[tex]S_{8}=\frac{8}{2} [2\times 17+(8-1)\times 1][/tex]
= 4(34 + 7)
= 164
Sum of 15 + [tex]S_{8}[/tex] = 15 + 164 = 179
Now the difference between 41 and sum of perfect squares between 15 and 25 inclusive = [tex]41-179[/tex]
= -138
Therefore, answer is (-138).
The sum of the perfect squares (16 and 25) between 15 and 25 is 41. The sum of all the other numbers between 15 and 25 is 179. So, when we subtract 179 from 41, we get -138.
Explanation:To find the answer, we first need to identify which numbers between 15 and 25, inclusive, are perfect squares. A perfect square is a number which can be expressed as the product of an integer with itself. In this range, 16 and 25 are the perfect squares.
The sum of these perfect squares is 16 + 25 = 41.
Next, we need to find the sum of all the other numbers between 15 and 25, inclusive, that are not perfect squares. These numbers are 15, 17, 18, 19, 20, 21, 22, 23, and 24. Adding these numbers together gives a sum of 179.
Now subtract this second sum from the first to get the answer: 41 - 179 = -138
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