Answer:
The age of Sleepy is 1 .
Then the age of Scoopy is 9.
The age of Doc would be 45.
Step-by-step explanation:
We have to find the ages of all the three people based on the given equations.
Let the age of Sleepy be x. Since Scoopy is 8 years older than Sleepy, the age of Scoopy would be (x+8).
Also Doc is 5 times as old as Scoopy so his age is 5(x+8).
Now the sum of the ages of all three is 55.
Writing equation for this,
x + x + 8 + 5(x+8) = 55
7x + 48 = 55
7x = 7
x = 1
So the age of Sleepy is 1 .Then the age of Scoopy is 9.
The age of Doc would be 45.
By solving the linear equations given by the problem, it is found that Scoopy is 10 years old, Sleepy is 2 years old, and Doc is 50 years old.
Explanation:This problem is a system of linear equations, a topic in the field of mathematics. The task is to find out the age of Sleepy, Scoopy, and Doc. Let's denote Scoopy's age as 'S', Sleepy's age as 'S - 8' (since Sleepy is eight years younger than Scoopy), and Doc's age as '5S' (since Doc is 5 times as old as Scoopy). The equations can be written as below:
Scoopy's age = SSleepy's age = S - 8Doc's age = 5SIt is known that the sum of their ages is 55, which gives us our second equation:
S + (S - 8) + 5S = 55
By solving this equation, we find that S = 10. So Scoopy is 10 years old. Accordingly, Sleepy is 10 - 8 = 2 years old and Doc is 5 * 10 = 50 years old.
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Last year the girls' basketball team had 8 fifth-grade students and 7 sixth-grade students. What was the ratio of sixth-grade students to fifth-grade students on the team?
The ratio of sixth-grade students to fifth-grade students is 7/8 or 7:8
Step-by-step explanation:
Let us define ratio first.
"Ratio is the quantitative relationship between two quantities which tells that one quantity is how many times of another quantity"
Given
Number of fifth grade students = 8
Number of sixth grade students = 7
We have to find the ratio of sixth grade students to fifth grade students
So,
[tex]r = \frac{sixth-grade\ students}{fifth-grade\ students}\\r = \frac{7}{8}\\r = 7:8[/tex]
Hence,
The ratio of sixth-grade students to fifth-grade students is 7/8 or 7:8
Keywords: ratio, fraction
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Malcolm is trying a very low-carbohydrate diet. He would like to keep the amount of carbs consumed in grams between the levels shown in the following compound inequality:
50 < 2x + 10 and 2x + 10 < 110
Solve for x in this inequality, and explain what the answer represents.
The solution to the compound inequality is 20 < x < 50. Malcolm should consume more than 20 but less than 50 grams of carbohydrates to adhere to his low-carbohydrate diet.
Explanation:To solve the compound inequality 50 < 2x + 10 and 2x + 10 < 110 for Malcolm's diet, we'll deal with each part of the inequality separately and then combine the results to find the range of values for x that satisfies both conditions.
For the first part, subtract 10 from both sides: 50 - 10 < 2x, which simplifies to 40 < 2x. Then, divide both sides by 2 to isolate x: 20 < x.For the second part, subtract 10 from both sides: 2x + 10 - 10 < 110 - 10, simplifying to 2x < 100. Dividing both sides by 2 yields: x < 50.Combining these inequalities gives us 20 < x < 50. This means that Malcolm should consume more than 20 but less than 50 grams of carbohydrates to stay within his targeted dietary range on his low-carbohydrate diet.
A garden hose emits 9 quarts of water in 6 seconds at this rate How long will it take the hose to emit 12 quarts how much water does the hose emit in 10 second
Answer:
8 seconds
15 quarts
Step-by-step explanation:
Givens
6 seconds emits 9 quarts of water.
x seconds emits 12 quarts of water.
Formula
6/x = 9/12
Solution
Cross multiply
9x = 12*6 Simplify the right
9x = 72 Divide by 9
9x/9 = 72/9
x = 8 seconds Answer.
In 8 seconds, the hose will emit 12 quarts
=========================
Givens
9 quarts of water are emitted in 6 second
x quarts of water are emitted in 10 seconds
Formula
9/x = 6/10 Cross multiply
Solution
9*10 = 6x Combine the left
90 = 6x Divide by 6
90/6=6x/6
15 = x
In 10 seconds the hose emits 15 quarts.
Answer:
Step-by-step explanation:
The garden hose emits 9 quarts of water in 6 seconds. It means that the number of seconds that it takes the garden hose to emit 1 quart of water would be 6/9 = 2/3 seconds
Therefore, the number of seconds that it will take the garden hose to emit 12 quarts of water would be
12 × 2/3 = 8 seconds.
Again, the number of quarts that the garden hose will emit in one second is 9/6 = 3/2 quarts.
Therefore, the number of quarts of water that the hose emits in 10 seconds would be
10 × 3/2 = 15 quarts.
Julia is allowed to watch no more than 5 hours of television a week. So far this week, she has watched 1.5 hours. Write and solve an inequality to show how many hours of television Julia can still watch this week.
The inequality is used to solve how many hours of television Julia can still watch this week is [tex]x + 1.5 \leq 5[/tex]
The remaining hours of TV Julia can watch this week can be expressed is 3.5 hours
Solution:Given that Julia is allowed to watch no more than 5 hours of television a week
So far this week, she has watched 1.5 hours
To find: number of hours Julia can still watch this week
Let "x" be the number of hours Julia can still watch television this week
"no more than 5" means less than or equal to 5 ( ≤ 5 )
Juila has already watched 1.5 hours. So we can add 1.5 hours and number of hours Julia can still watch television this week which is less than or equal to 5 hours
number of hours Julia can still watch television this week + already watched ≤ Total hours Juila can watch
[tex]x + 1.5 \leq5[/tex]
Thus the above inequality is used to solve how many hours of television Julia can still watch this week.
Solving the inequality,
[tex]x + 1.5 \leq5\\\\x \leq 5 - 1.5\\\\x \leq 3.5[/tex]
Thus Julia still can watch Television for 3.5 hours
Prove that it is impossible to dissect a cube into finitely many cubes, no two of which are the same size.
explanation:
The sides of a cube are squares, and they are covered by the respective sides of the cubes covering that side of the big cube. If we can show that a sqaure cannot be descomposed in squares of different sides, then we are done.
We cover the bottom side of that square with the bottom side of smaller squares. Above each square there is at least one square. Those squares have different heights, and they can have more or less (but not equal) height than the square they have below.
There is one square, lets call it A, that has minimum height among the squares that cover the bottom line, a bigger sqaure cannot fit above A because it would overlap with A's neighbours, so the selected square, lets call it B, should have less height than A itself.
There should be a 'hole' between B and at least one of A's neighbours, this hole is a rectangle with height equal to B's height. Since we cant use squares of similar sizes, we need at least 2 squares covering the 'hole', or a big sqaure that will form another hole above B, making this problem inifnite. If we use 2 or more squares, those sqaures height's combined should be at least equal than the height of B. Lets call C the small square that is next to B and above A in the 'hole'. C has even less height than B (otherwise, C would form the 'hole' above B as we described before). There are 2 possibilities:
C has similar size than the difference between A and BC has smaller size than the difference between A and BIf the second case would be true, next to C and above A there should be another 'hole', making this problem infinite. Assuming the first case is true, then C would fit perfectly above A and between B and A's neighborhood. Leaving a small rectangle above it that was part of the original hole.
That small rectangle has base length similar than the sides of C, so it cant be covered by a single square. The small sqaure you would use to cover that rectangle that is above to C and next to B, lets call it D, would leave another 'hole' above C and between D and A's neighborhood.
As you can see, this problem recursively forces you to use smaller and smaller squares, to a never end. You cant cover a sqaure with a finite number of squares and, as a result, you cant cover a cube with finite cubes.
(3 points)
11. A farmer buys 20 sheep, half male and half female. She was told that the annual rate of
increase for the sheep population is 60%. Assuming that none of the sheep die, when will the
farmer have 200 sheep? Write and solve an exponential equation, showing your work.
Use
to indicate an exponent. Use /to indicate a fraction.
Answer:
The exponential Function is [tex]20+12h=200[/tex].
Farmer will have 200 sheep after 15 years.
Step-by-step explanation:
Given:
Number of sheep bought = 20
Annual Rate of increase in sheep = 60%
We need to find that after how many years the farmer will have 200 sheep.
Let the number of years be 'h'
First we will find the Number of sheep increase in 1 year.
Number of sheep increase in 1 year is equal to Annual Rate of increase in sheep multiplied by Number of sheep bought and then divide by 100.
framing in equation form we get;
Number of sheep increase in 1 year = [tex]\frac{60}{100}\times20 = 12[/tex]
Now we know that the number of years farmer will have 200 sheep can be calculated by Number of sheep bought plus Number of sheep increase in 1 year multiplied by number of years is equal to 200.
Framing in equation form we get;
[tex]20+12h=200[/tex]
The exponential Function is [tex]20+12h=200[/tex].
Subtracting both side by 20 using subtraction property we get;
[tex]20+12h-20=200-20\\\\12h=180[/tex]
Now Dividing both side by 12 using Division property we get;
[tex]\frac{12h}{12} = \frac{180}{12}\\\\h =15[/tex]
Hence Farmer will have 200 sheep after 15 years.
Carter has 37 coins, all nickels and dimes in his piggy bank. The value of the coins is $3.10. How many dimes does carter have have. Use substitution or elimination method
Carter has 25 dimes
Solution:
Let "n" be the number of nickels
Let "d" be the number of dimes
Given that Carter has 37 coins, all nickels and dimes in his piggy bank
number of nickels + number of dimes = 37
n + d = 37 -------- eqn 1
Given that value of coins is $ 3.10
Also, value of nickel is 0.05 dollar and value of dime is 0.10 (in dollars) and total value of these 37 coins is 3.10, so we can write:
number of nickels x value of 1 nickel + number of dimes x value of 1 dime = 3.10
[tex]n \times 0.05 + d \times 0.10 = 3.10[/tex]
0.05n + 0.10d = 3.10 ------- eqn 2
Let us solve eqn 1 and eqn 2 to find values of "n" and "d"
From eqn 1,
n = 37 - d ---- eqn 3
Substitute eqn 3 in eqn 2
0.05(37 - d) + 0.10d = 3.10
1.85 - 0.05d + 0.10d = 3.10
0.05d = 3.10 - 1.85
0.05d = 1.25
d = 25Thus carter has 25 dimes
By setting up and solving a system of equations, we find that Carter has 25 dimes in his piggy bank.
Explanation:To solve how many dimes Carter has, we need to set up two equations based on the given information.
Let the number of nickels be n and the number of dimes be d.The total number of coins is 37, so n + d = 37.Each nickel is worth 5 cents and each dime is worth 10 cents, and the total value of the coins is $3.10 or 310 cents. So, the value equation is 5n + 10d = 310.Now, you can use the substitution or elimination method to solve the system of equations. Let's use substitution. From the first equation, we can express d as 37 - n.Substitute d in the value equation: 5n + 10(37 - n) = 310.Simplifying, 5n + 370 - 10n = 310, which gives us -5n = -60.Dividing by -5, we find n = 12.Substitute n back into d = 37 - n to find d = 37 - 12 = 25.Carter has 25 dimes in his piggy bank.
During the 31-day month of May, a tuxedo shop rents a different number of tuxedos each day, including a store-record 55 tuxedos on May 23rd. Assuming that the shop had an unlimited inventory of tuxedos to rent, what is the maximum number of tuxedos the shop could have rented during May?
Answer:
The answer is 1240
Step-by-step explanation:
We know that tuxedo shop rents different number of tuxedos each day and at max they rented 55 tuxedos in May. To maximize the number of tuxedos that rented, rented number of tuxedos need to be drop down by 1 for each day. So minimum 25 tuxedos were rented in May. To find the maximum number of tuxedos that rented in May, we need to add all numbers with using the formula for adding consecutive numbers.
[tex]Total = ((55+25)/2)*31=1240[/tex]
Sydney is cutting the crust from the edges of her sandwich. The dimensions, in centimeters, of the sandwich is shown.
A rectangle labeled sandwich. The right side is labeled 2 x squared + 9. The bottom side is labeled 2 x squared + 8.
Which expression represents the total perimeter of her sandwich, and if x = 1.2, what is the approximate length of the crust?
8x2 + 34; 43.6 centimeters
8x2 + 34; 45.52 centimeters
4x2 + 17; 21.8 centimeters
4x2 + 17; 22.76 centimeters
Answer:
8x^2 + 34; 45.52 centimeters
Step-by-step explanation:
The perimeter is twice the sum of adjacent edges of a rectangle:
P = 2((2x^2 +9) +(2x^2 +8)) = 2(4x^2 +17)
P = 8x^2 +34
Then when x=1.2, the perimeter is ...
P = 8(1.2^2) +34 = 8·1.44 +34 = 45.52 . . . . cm
Answer:
The Answer is: B 8x^2 + 34; 45.52 centimeters
Step-by-step explanation:
Took it on edg
PLEASE HELP
CD¯ has endpoints C and D, with C at coordinates (5,8). CD¯- has midpoint M at (3,9).
What are the coordinates of point D?
Answer:
(1, 10)
Step-by-step explanation:
Use midpoint formula.
Mₓ = (x₁ + x₂) / 2
3 = (5 + x) / 2
6 = 5 + x
x = 1
Mᵧ = (y₁ + y₂) / 2
9 = (8 + y) / 2
18 = 8 + y
y = 10
The coordinates of point D are (1, 10).
The coordinates of point D are (1, 10).
What does a midpoint mean?Midpoint, as the word suggests, means the point which lies in the middle of something.
Midpoint of a line segment means a point which lies in the mid of the given line segment.
Given; CD has endpoints C and D, with C at coordinates (5,8). CD has midpoint M at (3,9).
Using midpoint formula;
Mₓ = (x₁ + x₂) / 2
3 = (5 + x) / 2
6 = 5 + x
x = 1
Now, Mᵧ = (y₁ + y₂) / 2
9 = (8 + y) / 2
18 = 8 + y
y = 10
Hence, The coordinates of point D are (1, 10).
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A family has two cars. During one particular week, the first car consumed 25 gallons of gas and the second consumed 40 gallons of gas. The two cars drove a combined total of 1225 miles, and the sum of their fuel efficiencies was 40 miles per gallon. What were the fuel efficiencies of each of the cars that week?
First Car: ~ miles per gallon
Second Car: ~ miles per gallon
Answer:
first car: 25 mpgsecond car: 15 mpgStep-by-step explanation:
Let x represent the fuel efficiency of the car that used the most gas. Since the sum of the fuel efficiencies is 40 mpg, the other car had a fuel efficiency of (40-x). Then the combined miles driven is ...
40x +25(40-x) = 1225
15x = 225
x = 15
(40-x) = 25
The first car got 25 miles per gallon; the second car got 15 miles per gallon.
_____
We made use of the relation ...
gallons × (miles/gallon) = miles
A rectangular poster is 3 times as long as it is wide. A rectangular banner is 5 times as long as it is wide. Both the banner and the poster have perimeters of 24 inches. What are the lengths and wides of the poster and the poster?
Answer:
Length of poster is 9 inches and width of the poster is 3 inches.
Length of banner is 10 inches and width of the banner is 2 inches.
Step-by-step explanation:
Given:
Perimeter of Banner =24 in.
Perimeter of poster = 24 in.
we need to find the dimensions of poster and banner.
First we will find the dimension of poster.
Now Given:
A rectangular poster is 3 times as long as it is wide.
Let the Width of poster be [tex]'p'[/tex].
Length of the poster = [tex]3p[/tex]
Perimeter of poster = 24 in.
But perimeter of poster is equal to twice the sum of length and width.
framing in equation form we get;
[tex]2(p+3p)=24\\\\2(4p)=24\\\\8p=24\\\\p=\frac{24}{8} = 3\ in.[/tex]
Now width of poster = 3 inches
Length of the poster = [tex]3p = 3\times3 =9\ inches[/tex]
Hence Length of poster is 9 inches and width of the poster is 3 inches.
Now we will find the dimension of Banner.
Now Given:
A rectangular banner is 5 times as long as it is wide.
Let the Width of Banner be [tex]'b'[/tex].
Length of the banner = [tex]5b[/tex]
Perimeter of banner = 24 in.
But perimeter of banner is equal to twice the sum of length and width.
framing in equation form we get;
[tex]2(b+5b)=24\\\\2(6b)=24\\\\12b=24\\\\b=\frac{24}{12} = 2\ in.[/tex]
Now width of banner = 2 inches
Length of the banner = [tex]5b = 5\times2 = 10\ inches[/tex]
Hence Length of banner is 10 inches and width of the banner is 2 inches.
The width and length of the rectangular poster are 3 inches and 9 inches, respectively. The width and length of the rectangular banner are 2 inches and 10 inches, respectively. Both were calculated by setting up equations using the perimeter formula for rectangles.
Explanation:The problem is to determine the lengths and widths of a rectangular poster and a rectangular banner, both having the same perimeter of 24 inches. The poster's length is 3 times its width, while the banner's length is 5 times its width. We'll set up two separate equations for their perimeters and solve for the width and length of each.
Poster:Let w be the width of the poster. Then the length is 3w. The perimeter is given by P = 2l + 2w, where P is the perimeter and l is the length. This gives us:
24 = 2(3w) + 2w -> 24 = 6w + 2w -> 24 = 8w -> w = 3 inches
Therefore, the length of the poster is 3w = 9 inches.
Banner:Let x be the width of the banner. Then the length is 5x. Again using the perimeter formula, we get:
24 = 2(5x) + 2x -> 24 = 10x + 2x -> 24 = 12x -> x = 2 inches
Therefore, the length of the banner is 5x = 10 inches.
Suppose that $p$ and $q$ are positive numbers for which \[\log_9 p = \log_{12} q = \log_{16} (p + q).\] Then $q/p$ can be expressed in the form $(x + \sqrt{y})/z$, where $x$, $y$, and $z$ are positive integers, and $y$ is not divisible by the square of a prime. Find $x + y + z$.
Value of (x + y + z) = 8
Suppose p and q are the positive numbers for which
[tex]log_(9)p=log_(12)q=log_(16)(p+q)[/tex] from the given expression,
[tex]log_(9)p=log_(12)q[/tex]
[tex](logp)/(log9)=(log(q))/(log12)[/tex]
log(p).log(12) = log(q).log(9)
log(q).2log(3) = log(p).log(12) ------(1)
[tex]Now log_(12)q=log_(16)(p+q)[/tex]
[tex](logq)/(log12)=(log(p+q))/(log(16))[/tex]
log(q).log(16) = log(p + q).log(12)
2log(4).log(q) = log(p + q).log12 -------(2)
By adding both the equations (1) and (2),
2log(3).log(q) + 2log(4).log(q) = log(12).log(p) + log(12).log(p + q)
log(q)[2log(3) + 2log(4)] = log(12)[logp + log(p + q)]
2log(q).log(12) = log(12).log[p.(p + q)]
2log(q) = log[p.(p+q)]
q² = p(p + q)
(q)/(p)=(p+q)/(q)
(q)/(p)=(p)/(q)+1
Let (q)/(p)=a
a = (1)/(a)+1
a² - a - 1 = 0 from quadratic formula,
a = [tex]\frac{1\pm \sqrt{(-1)^(2)-4 \times 1 \times (-1)}}{2}[/tex]
a = [tex](1\pm √((1+4)))/(2)[/tex]
a = [tex](1\pm √((5)))/(2)[/tex]
If the solution is represented by (x+√(y))/(z) then it will be equal to
(1+√((5)))/(2) then x = 1, y = 5 and z = 2.
Now we have to find the value of (x + y + z).
By placing the values of x, y and z,
(x + y + z) = (1 + 5 + 2) = 8
Therefore, value of (x + y + z) = 8
The price of a certain painting increased by 20% during the first year and decreased by 15% during the second year. The price of the painting at the end of the 2-year period was what percent of the original price?
A. 102%B. 105%C. 120%D. 135%E. 140%
Answer:
A.102%
Step-by-step explanation:
Let cost price of painting=$100
In first year price increased 20%
Then , the price=[tex]100+100(0.20)=[/tex]$120
In second year
Price decreased 15%
Then , the price of painting=[tex]120-120(0.15)[/tex]
The price of painting=$102
Percent =[tex]\frac{final\;price}{Initial\;price}\times 100[/tex]
By using this formula
Then, we get
Percent of the original price=[tex]\frac{102}{100}\times 100[/tex]
Percent of the original price=102%
Option A is true.
Each of the integers $1,$ $2,$ $3,$ $\dots,$ $16$ is written on a separate slip of paper and these slips are placed in a pile. Jillian will randomly draw slips from the pile without replacement and will continue drawing until two of the numbers she has drawn from the pile have a product that is a perfect square. What is the maximum number of slips that Jillian can draw without obtaining a product that is a perfect square?
Answer:
11 slips
Step-by-step explanation:
A perfect square is a positive integer that is the square of another integer. For example, 25 is a perfect square of 5. See the calculation below
.
[tex]5^{2} = 5*5\\5^{2} = 25[/tex]
There is a rule that should be kept in mind
1. When two perfect squares are multiplied by each other (e.g. 4 * 9), the result is a perfect square ([tex]36 = 6^{2}[/tex])
Let's identify the combination of numbers that result in perfect square, when multiplied with each other. These combinations are as follows,
• 1*4, 1*9, 1*16
• 2*8
• 3*12
• 4*9, 4*16
• 9*16
From the list of numbers 1, 4, 9 and 16 are already perfect square e.g. [tex]2^{2} = 4, 3^{2} = 9[/tex]. If they are multiplied by each other, the result will also be a perfect square. Let’s assume that our first number is 1. Now we can't have any of the three numbers (except for 1), mentioned above. This rule out these three numbers.
Next, from 2, 8, 3 and 12 we can only draw two numbers. e.g. if we take 2, we can’t take 8 as it will give a perfect square. Same goes with 3 and 12. Hence from these four numbers we can discard two of them.
We discarded three numbers initially and two now. Therefore, out of 16 slipds we can draw a maximum of 11 slips without obtaining a product that is a perfect square.
slader The magnitude of earthquakes recorded in a region of North America can be modeled ashaving an exponential distribution with mean 2.4, as measured on the Richter scale. Find theprobability that an earthquake striking this region will a exceed 3.0 on the Richter scale. b fall between 2.0 and 3.0 on the Richter scale.
Answer:
a) [tex]P(X>3.0)=1-P(X \leq 3.0)= 1- [1- e^{-\frac{1}{2.4} 3.0}]=e^{-\frac{1}{2.4} 3.0}=0.287[/tex]
b) [tex]P(2<X<3)= 1- e^{-\frac{1}{2.4} 3} -[ 1- e^{-\frac{1}{2.4} 2}][/tex]
[tex]P(2<X<3)=e^{-\frac{1}{2.4} 2}- e^{-\frac{1}{2.4} 3}=0.435-0.287=0.148[/tex]
Step-by-step explanation:
Definitions and concepts
The Poisson process is useful when we want to analyze the probability of ocurrence of an event in a time specified. The probability distribution for a random variable X following the Poisson distribution is given by:
[tex]P(X=x) =\lambda^x \frac{e^{-\lambda}}{x!}[/tex]
And the parameter [tex]\lambda[/tex] represent the average ocurrence rate per unit of time.
The exponential distribution is useful when we want to describ the waiting time between Poisson occurrences. If we assume that the random variable T represent the waiting time btween two consecutive event, we can define the probability that 0 events occurs between the start and a time t, like this:
[tex]P(X>x)= e^{-\lambda x}[/tex]
We can express in terms of the mean [tex]\mu =\frac{1}{\lambda}[/tex]
[tex]P(X>x)= e^{-\frac{1}{\mu} x}[/tex]
And the cumulative function would be given by the complement rule like this:
[tex]P(X\leq x)=1- e^{-\frac{1}{\mu} x}[/tex]
Solution for the problem
For this case we have that X the random variable that represent the magnitude of earthquakes recorded in a region of North America, we know that the distribution is given by:
[tex]X\sim Expon(\mu =2.4)[/tex]
a) Find theprobability that an earthquake striking this region will a exceed 3.0 on the Richter scale
So for this case we want this probability:
[tex]P(X>3.0)=1-P(X \leq 3.0)= 1- [1- e^{-\frac{1}{2.4} 3.0}]=e^{-\frac{1}{2.4} 3.0}=0.287[/tex]
b) fall between 2.0 and 3.0 on the Richter scale.
For this case we want this probability:
[tex]P(2 < X <3) = P(X<3) -P(X<2)[/tex]
And replacing we have this:
[tex]P(2<X<3)= 1- e^{-\frac{1}{2.4} 3} -[ 1- e^{-\frac{1}{2.4} 2}][/tex]
[tex]P(2<X<3)=e^{-\frac{1}{2.4} 2}- e^{-\frac{1}{2.4} 3}=0.435-0.287=0.148[/tex]
Simon has 20 quarters and 12 dimes.He wants to purchase ice cream for his friends.An ice cream cone cost $1.00.How many cones can Simon buy for his friends?
Answer:
6
Step-by-step explanation:
Each ice cream cone costs 4 quarters or 10 dimes, so Simon has ...
20/4 + 12/10 = 5 + 1.2 = 6.2
times the price of an ice cream cone. He can buy 6 cones for his friends.
Write a correct equation for the function whose graph is shown.
Answer:
[tex]\displaystyle y=-2sin(6t)[/tex]
Step-by-step explanation:
Sinusoid
The shape of a sinusoid is well-know because it describes a curve with a smooth and periodic oscillation. The sine and cosine are the two trigonometric functions in which the graph matches the description of a sinusoid. The sine can be identified because its value is zero at time zero.
The graph shown in the figure corresponds to a sine. The other characteristics of the sine function are
* It completes a cycle in[tex]2\pi[/tex] radians
* It has a maximum of 1 and a minimum of -1
* It's increasing for a quarter of the cycle, decreasing for half of the cycle, and increasing for the remaining quarter of the cycle
* The equation is
[tex]y=Asin(wt)[/tex]
The function starts decreasing for the first quarter, which only is possible if the amplitude A is negative. We can also see the maximum and minimum values are 2 and -2 respectively. This means the amplitude is A=-2
We can also see the function completes 3 cycles in [tex]t=\pi[/tex] radians or 6 cycles in [tex]2\pi[/tex] radians. Or, equivalently
[tex]wt=12\pi[/tex]
[tex]w(2\pi)=12\pi[/tex]
[tex]\displaystyle w=\frac{12}{2}=6\ rad/sec[/tex]
Thus, the function can be expressed as
[tex]\boxed{\displaystyle y=-2sin(6t)}[/tex]
Answer:
y=-2sin6x
Step-by-step explanation:
Consider a roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. If Smith always bets that the outcome will be one of the numbers 1 through 12,
what is the probability that
a. Smith will lose his first 5 bets;
b. his first win will occur on his fourth bet?
Answer:
The probability that Smith will lose his first 5 bets is 0.15
The probability that his first win will occur on his fourth bet is 0.1012
Step-by-step explanation:
Consider the provided information.
A roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. Smith always bets that the outcome will be one of the numbers 1 through 12,
It is given that Smith always bets on the numbers 1 through 12.
There are 12 numbers from 1 to 12.
Thus, the probability of success (winning) is= [tex]\frac{12}{38}[/tex]
The probability of not success (loses) is= [tex]1-\frac{12}{38}=\frac{26}{38}[/tex]
Part (A) Smith will lose his first 5 bets.
The probability that Smith loses his first 5 bets is,
[tex]\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}=(\frac{26}{38})^5\approx0.15[/tex]
Hence, the probability that Smith will lose his first 5 bets is 0.15
Part (B) His first win will occur on his fourth bet?
Smith’s first win occurring on the fourth bet means that he loses the first 3 bets and wins on the fourth bet. That is,
[tex]\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{12}{38}=(\frac{26}{38})^3\times\frac{12}{38}\approx0.1012[/tex]
Hence, the probability that his first win will occur on his fourth bet is 0.1012
Sam needs to make a long-distance call from apay phone.With his prepaid phone card, he will be charged $1.00 to connect and $0.50 per minute.If he places a collect call with the operator he will be charged $3.00 to connect and $0.25 per minute.In how many minutes will the phone card and the operator-assisted call cost the same?
A.5 minutes
B. 51/3 min.
C. 8 min.
D. 16 min.
Answer:
The answer is 8
Step-by-step explanation:
Lets consider that Sam's call will last x minutes in total. Then, equations for cost will be as follows:
A Pay Phone Cost: [tex]p(x)=1+0,5*x[/tex]
Operator Cost: [tex]o(x)=3+0,25*x[/tex]
If we make both costs equal, the equation will be:
[tex]1+0,5*x=3+0,25*x\\0,25*x=2\\x=8[/tex]
Sam's call need to last 8 minutes to make payphone cost and operator cost same.
Vector u has its initial point at (21, 12) and its terminal point at (19, -8). Vector v has a direction opposite that of u, whose magnitude is five times the magnitude of v. Which is the correct form of vector v expressed as a linear combination of the unit vectors i and j?
[tex]\boxed{\vec{v}=\frac{2}{5}i+4j}[/tex]
Explanation:In this exercise, we have the following facts for the vector [tex]\vec{u}[/tex]:
It has its initial point at [tex](21,12)[/tex], let's call it [tex]P_{1}[/tex] It has its terminal point at [tex](19,-8)[/tex], let's call it [tex]P_{2}[/tex]Since the vector [tex]\vec{u}[/tex] goes from point [tex]P_{1}[/tex] to [tex]P_{2}[/tex], then:
[tex]\vec{u}=(19,-8)-(21,12) \\ \\ \vec{u}=(19-21,-8-12) \\ \\ \vec{u}=(-2,-20)[/tex]
On the other hand, we have the following facts for the vector [tex]\vec{v}[/tex]:
Vector [tex]\vec{v}[/tex] has a direction opposite that of [tex]\vec{u}[/tex], The magnitude of [tex]\vec{u}[/tex] is five times the magnitude of [tex]v[/tex].So we can write this relationship as follows:
[tex]5\vec{v}=-\vec{u} \\ \\ \vec{v}=-\frac{1}{5}\vec{u} \\ \\ \vec{v}=-\frac{1}{5}(-2,-20) \\ \\ \vec{v}=(\frac{2}{5},4) \\ \\ \\ Finally: \\ \\ \boxed{\vec{v}=\frac{2}{5}i+4j}[/tex]
Learn more:Length of vectors: https://brainly.com/question/12264340
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Cousin Edie drank all of Clark's egg nog.Edie finds a coupon for 50 cents off that can only be use at a local grocery store.Clark normally buys his egg nog at the supermarket. The tax rate at both stores is 2.25% .Which deal is better? Clark:$5:95 plus tax Eddie:6.35+coupon tax
Answer:
Edie has a better deal since his final price is lower
Step-by-step explanation:
Cousin Edie drank all of Clark's egg nog . Edie finds a coupon for 50 cents off that can only be use at a local grocery store.Clark normally buys his egg nog at the supermarket.
The tax rate at both stores is 2.25%.
Clark buys the egg nog at $5.95 plus tax .
Edie buys the egg nog at $6.35 plus coupon plus tax.
After Edie applies the coupon , the final price is $( 6.35 - 0.50 ) = $5.85
Since the percentage of tax applied is the same, the deal with lower final price is better.
Hence, Edie has a better deal.
For the trapezoid, ABCD E and F are the midpoints of
AC
and
BD
respectively.
Find the length of the segment
EF
, if AB = a, BC = b, CD = c, AD = d.
Answer:
EF = (d − b) / 2
Step-by-step explanation:
Let's say that G is the intersection of the trapezoid's diagonals.
Triangle GBC is similar to triangle GDA, so we can write a proportion:
GD / GB = AD / BC
GD / GB = d / b
GD = (d / b) GB
Next, F is the midpoint of BD, so BF equals FD.
BF = FD
GB + GF = GD − GF
2GF = GD − GB
2GF = (d / b) GB − GB
2GF = ((d − b) / b) GB
GF / GB = (d − b) / (2b)
Finally, triangle GBC is similar to triangle GFE, so we can write another proportion:
GF / GB = EF / BC
(d − b) / (2b) = EF / b
EF = (d − b) / 2
The number of states that joined the United States between 1776 and 1849 is twice the number of states that joined between 1850 and 1900. If 30 states joined the United States between 1776and 1849 and x states joined between 1850 and 1900, which of the following equations is true?
a. 30x = 2
b. 2x = 30
c. x/2 = 30
d. x + 30 = 2
Answer: b. 2x = 30
Step-by-step explanation:
Given : 30 states joined the United States between 1776 and 1849
and x states joined between 1850 and 1900 .
If the number of states that joined the United States between 1776 and 1849 is twice the number of states that joined between 1850 and 1900.
i.e. No. of states joined the United States between 1776 and 1849= 2 (No. of states that joined between 1850 and 1900)
i.e . 30= 2(x) [Substitute the values]
i.e . 2x=30
Hence, the true equation : 2x=30
A differential equation that is a function of y only
a.will produce a slope field with parallel tangents along the diagonal
b.will produce a slope field that does not have rows or columns of parallel tangents
c.will produce a slope field with rows of parallel tangents
d.will produce a slope field with columns of parallel tangents
Answer:
c. Will produce a slope field with rows of parallel tangents
Step-by-step explanation:
We can write a differential equation that is a function of y only as:
[tex]y'=f(y)[/tex]
So the derivative, in this particular case, of any function will be a function of the dependent variable y only, it means that the curves you will get should all be pointing in the same direction for each value of x. Therefore the sketch of the slopes field would have parallel curves for each value of x, in other words, it will produce a slope field with rows of parallel tangents.
I hope it helps you!
Of the 645 speckled trout in a certain fishery that contains only speckled and rainbow trout, the number of males is 45 more than twice the number of females. If the ratio of female speckled trout to male rainbow trout is 4:3 and the ratio of male rainbow trout to all trout is 3:20, how many female rainbow trout are there?A. 192B. 195C. 200D. 205E. 208
Answer: D 205
Step-by-step explanation:
Let,
Number of all trouts = N
Number of speckled trouts = Ns = 645
Number of rainbow trouts = Nr
Number of male speckled trouts = Ms
Number of female speckled trouts = Fs
Number of male rainbow trouts = Mr
Number of female rainbow trouts = Fr
Since, Ms = 2Fs + 45
Also, Ms + Fs = 645
Therefore, 2Fs + 45 + Fs = 645
Fs = (645-45)/3 = 200
Female speckled trouts = 200
Since
Ms + Fs = 645
Ms = 645- 200 = 445
Since, Fs/Mr = 4/3
Mr = 3x200/4 = 150
Since,
Mr/N = 3/20
N = 20x 150/3 = 1000
Recall that,
N = Ms+Fs+Mr+Fr
Fr = N-Ms-Fs-Mr
Fr = 1000-445-200-150
Fr = 205
Therefore, the number of rainbow female trouts = 205
The number of female rainbow trout is calculated using given ratios and the total number of speckled trout. After solving a series of equations, the answer is determined to be 205 female rainbow trout, which is option D.
To find the number of female rainbow trout, we first need to unpack the information provided in the question and express it in equations.
Let n be the number of female speckled trout. We know that the number of male speckled trout is 45 + 2n. Since there are 645 speckled trout in total, we can express this as:
n + (45 + 2n) = 645
Solving for n, we get:
3n + 45 = 645
3n = 600
n = 200
Now, we have the ratio of female speckled trout to male rainbow trout as 4:3, and since we've found there are 200 female speckled trout (n = 200), we can figure out the number of male rainbow trout. Let's call this number m. We have:
200/4 = m/3
50 = m/3
m = 150
The ratio of male rainbow trout to all trout is given as 3:20. If the total number of trout is T, then:
150/T = 3/20
We can solve for T:
20 × 150 = 3T
3000 = 3T
T = 1000
So there are 1000 trout in total, of which 645 are speckled. This means there must be 1000 - 645 = 355 rainbow trout. As we have already found there are 150 male rainbow trout, the remainder must be female. So:
355 - 150 = 205 female rainbow trout
Therefore, the correct answer is D. 205.
Mr. Jones took a survey of college students and found that 60 out of 65 students are liberal arts majors. If a college has 8,943 students, what is the expected number of students who are liberal arts majors? Answer quick
Answer:
We can expect 8255 numbers of students are liberal arts majors.
Step-by-step explanation:
Given:
Total Number of students in the college = 8943
Now According to Mr. Jones Survey;
60 out of 65 students are liberal arts majors.
We need to find the number of students who are liberal arts majors out of total number of students in college.
Solution:
First we will find the Percentage number of students who are liberal arts majors according to survey.
Percentage number of students who are liberal arts majors can be calculated by dividing 60 from 65 then multiplying by 100.
framing above quote in equation form we get;
Percentage number of students who are liberal arts majors = [tex]\frac{60}{65}\times 100 = 92.31\%[/tex]
Now we will find the Total number of students who are liberal arts major.
Total number of students who are liberal arts major can be calculated by Multiplying Percentage number of students who are liberal arts majors with total number of students in the college and then dividing by 100.
framing above quote in equation form we get;
Total number of students who are liberal arts major = [tex]\frac{92.31}{100}\times 8493 \approx8255.28[/tex]
Since number of students cannot be in point, so we will round the value.
Hence We can expect 8255 numbers of students are liberal arts majors.
Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.
Answer:
See proof below
Step-by-step explanation:
Let [tex]r=\frac{w+z}{1+wz}[/tex]. If w=-z, then r=0 and r is real. Suppose that w≠-z, that is, r≠0.
Remember this useful identity: if x is a complex number then [tex]x\bar{x}=|x|^2[/tex] where [tex]\bar{x}[/tex] is the conjugate of x.
Now, using the properties of the conjugate (the conjugate of the sum(product) of two numbers is the sum(product) of the conjugates):
[tex]\frac{r}{\bar{r}}=\frac{w+z}{1+wz} \left(\frac{1+\bar{w}\bar{z}}{\bar{w}+\bar{z}}{\right)[/tex]
=[tex]\frac{(w+z)(1+\bar{w}\bar{z})}{(1+wz)(\bar{w}+\bar{z})}=\frac{w+z+w\bar{w}\bar{z}+z\bar{z}\bar{w}}{\bar{w}+\bar{z}+\bar{w}wz+\bar{z}zw}=\frac{w+z+w+|w|^2\bar{z}+|z|^2\bar{w}}{\bar{w}+\bar{z}+|w|^2z+|z|^2w}=\frac{w+z+\bar{z}+\bar{w}}{\bar{w}+\bar{z}+z+w}=1[/tex]
Thus [tex]\frac{r}{\bar{r}}=1[/tex]. From this, [tex]r=\bar{r}[/tex]. A complex number is real if and only if it is equal to its conjugate, therefore r is real.
Last year, sales at a book store increased from $5,000 to $10,000. This year, sales decreased to $5,000 from $10,000. What percentage did sales increase last year? What percentage did sales decrease this year? Sales increased last year, from $5,000 to $10,000. When sales dropped from $10,000 to $5,000 this year, sales decreased .
Answer:
Step-by-step explanation:
Last year, sales at a book store increased from $5,000 to $10,000. The amount by which it increased would be 10000 - 5000 = $5000
The percentage by which the sales increased would be
5000/5000 × 100 = 100%
This year, sales decreased to $5,000 from $10,000.The amount by which it decreased would be 5000 - 10000 = - $5000
The percentage by which the sales increased would be
5000/10000 × 100 = 50%
Find an exact value.
sine of negative eleven pi divided by twelve.
quantity square root of six plus square root of two divided by four.
quantity negative square root of six minus square root of two divided by four.
quantity square root of two minus square root of six divided by four.
quantity square root of six minus square root of two divided by four.
Answer:
(√2 − √6) / 4
Step-by-step explanation:
Rewrite using special angles.
sin(-11π/12)
sin((4−15)π/12)
sin(4π/12 − 15π/12)
sin(π/3 − 5π/4)
Use angle difference formula:
sin(π/3) cos(5π/4) − sin(5π/4) cos(π/3)
Evaluate:
(√3/2) (-√2/2) − (-√2/2) (1/2)
-√6/4 + √2/4
(√2 − √6) / 4