Answer:
A: 6 and 24
B: 4 times as great; the rate of change increases exponentially
Step-by-step explanation:
Part A: The average rate of change on the interval [a, b] is given by ...
average rate of change = (h(b) -h(a))/(b -a)
On the interval [1, 2], the rate of change is ...
(h(2) -h(1))/(2 -1) = (12 -6)/1 = 6
On the interval [3, 4], the rate of change is ...
(h(4) -h(3))/(4 -3) = (48 -24)/1 = 24
For Section A, the average rate of change is 6; for Section B, the average rate of change is 24.
__
Part B: The ratio of the rates of change on the two intervals is ...
(RoC on [3,4]) / (RoC on [1,2]) = 24/6 = 4
The average rate of change of Section B is 4 times that of Section A.
__
The rate of change is exponentially increasing, so an interval of the same width that starts at "d" units more than the previous one will have a rate of change that is 2^d times as much.
A tank in the shape of a right circular cone has height 12 feet and base radius 4 feet. The tank is inverted, with its vertex pointing down and base at the top. The tank contains a liquid with weight density 63 pounds per cubic foot, but is filled to a depth of 8 feet only. Set up, but DO NOT EVALUATE, an integral for the work (in foot-pounds) required to pump all the liquid present to a height one foot over the top of the tank.
Answer:
V = int(π(y/3)^2, 0, 8)
(Definite integration of π(y/3)^2 with lower boundary 0 and upper boundary 8)
Step-by-step explanation:
Set up cartesian axis (x and y) to the system.
Let y axis as the line of the centre of the cone, passing through its vertex and the centre of it's circular base. The x axis could be the 90 degree line to the y axis that passes the vertex. So the origin (0,0) is at the vertex.
I'm this setup , looking it as if we are looking it in 2 dimension, we'll see that there is a signature straight line on the x-y plane, which this line will form the cone as it revolute around the y-axis
Find the equation of the line:
Using height 12 and radius base 4, we can get the slope of the line
m = 12/4 = 3
It passes through origin, so the y-intercept is 0
Hence, y = 3x
Since the volume revolves around y-axis, we use the equation volume of revolution around y-axis
V = int(πx^2,a,b)
(Definite integration of πx^2 with lower boundary a and upper boundary b
Since y=3x
x = y/3
For this que
V = int(π(y/3)^2, 0, 8)
(Definite integration of π(y/3)^2 with lower boundary 0 and upper boundary 8)
. Let A = (−2, 4) and B = (7, 6). Find the point P on the line y = 2 that makes the total distance AP + BP as small as possible.
Answer:
P(1,2)
Step-by-step explanation:
There are 2 points.
A(-2,4) and B(7,6)
the point P on the y=2 can also represented as P(x,2)
We can use the distance formula to find the distances AP and BP
[tex]\text{dist} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
for AP: A(-2,4) and P(x,2)
[tex]AP = \sqrt{(-2 - x)^2 + (4 - 2)^2}[/tex]
[tex]AP = \sqrt{(-2 - x)^2 + 4}[/tex]
[tex]AP = \sqrt{(-1)^2(2 + x)^2 + 4}[/tex]
[tex]AP = \sqrt{(2 + x)^2 + 4}[/tex]
for BP: B(7,6) and P(x,2)
[tex]BP = \sqrt{(7 - x)^2 + (6 - 2)^2}[/tex]
[tex]BP = \sqrt{(7 - x)^2 + 16}[/tex]
the total distance AP + BP will be
[tex]\sqrt{(2 + x)^2 + 4}+\sqrt{(7 - x)^2 + 16}[/tex] (plot is given below)
Our task is to find the value of x such that the above expression is small as possible. (we can find this either through plotting or differentiating)
If you plot the above equation, the minimum point of the curve will be clearly visible, and it will be at x = 1. Hence, the point P(1,2) is such that the total distance AP + BP is as small as possible.
The point P that makes the total distance AP + BP smallest on the line y=2 is given by the x-coordinate of the midpoint of A and B because the shortest distance is in a straight line. Therefore, the point P is (2.5, 2).
Explanation:To find the point P on the line y = 2 that makes the total distance AP + BP the smallest, you need to recall that the shortest distance between two points is a straight line. So, ideally, we want to find a point P (x,2) that is on the same vertical line (or x-coordinate) that intersects the line AB at the midpoint.
Step 1: Find the midpoint of A and B. The midpoint M is obtained by averaging the x and y coordinates of A and B: M = ((-2+7)/2 , (4+6)/2) = (2.5, 5).
Step 2: Since line y = 2 is horizontal, the x-coordinate of our point P will stay the same with the midpoint x-coordinate. Therefore, P has coordinates (2.5, 2).
So, the point on the line y = 2 that makes the total distance AP + BP as small as possible is P (2.5, 2).
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On the first day a total of 40 items were sold for $356. Define the variables and write a system of equations to find the number of cakes and pies sold
On the first day, a total of 40 items were sold for $356. Pies cost $10 and cakes cost $8. Define the variables, write a system of equations to find the number of cakes and pies sold, and state how many pies were sold.
Answer:The variables are defined as:
"c" represent the number of cakes sold and "p" represent the number of pies sold
The system of equations used are:
c + p = 40 and 8c + 10p = 356
18 pies and 22 cakes were sold
Solution:Let "c" represent the number of cakes sold
Let "p" represent the number of pies sold
Cost of 1 pie = $ 10
Cost of 1 cake = $ 8
Given that total of 40 items were sold
number of cakes + number of pies = 40
c + p = 40 ------ eqn 1
Given items were sold for $356
number of cakes sold x Cost of 1 cake + number of pies sold x Cost of 1 cake = 356
[tex]c \times 8 + p \times 10 = 356[/tex]
8c + 10p = 356 ----- eqn 2
Let us solve eqn 1 and eqn 2
From eqn 1,
p = 40 - c ---- eqn 3
Substitute eqn 3 in eqn 2
8c + 10(40 - c) = 356
8c + 400 - 10c = 356
-2c = - 44
c = 22
Substitute c = 22 in eqn 3
p = 40 - c
p = 40 - 22
p = 18
Thus 18 pies and 22 cakes were sold
In a pot worth $2.35, there are 6 quarters, 5 dimes, 5 pennies, and the rest of the coins are nickels. What is the ratio of nickels to dimes?
Answer:
6:5
Step-by-step explanation:
It is given that a pot worth $2.35 and there are 6 quarters, 5 dimes, 5 pennies, the rest of the coins are nickels.
We know that
$1 = 100 cents
1 penny = 1 cent = $0.01
1 nickel = 5 cents. = $0.05
1 dime = 10 cents. = $0.10
1 quarter = 25 cents = $0.25
The value of 6 quarters is
[tex]6\times 0.25=1.50[/tex]
The value of 5 dimes is
[tex]5\times 0.10=0.50[/tex]
The value of 5 pennies is
[tex]5\times 0.01=0.05[/tex]
Let x be the number of nickels. So, the value of x nickels is
[tex]x\times 0.05=0.05x[/tex]
Total value of 6 quarters, 5 dimes, 5 pennies, and x nickels is
[tex]Total =1.50+0.50+0.05+0.05x[/tex]
[tex]Total =2.05+0.05x[/tex]
It is given that the pot worth is $2.35.
[tex]2.05+0.05x=2.35[/tex]
Subtract 2.05 from both sides.
[tex]0.05x=0.30[/tex]
Divide both sides by 0.05.
[tex]x=6[/tex]
The number of nickels is 5.
[tex]\dfrac{Nickel}{Dimes}=\dfrac{6}{5}=6:5[/tex]
Therefore, the ratio of nickels to dimes is 6:5.
In a pot worth $2.35 containing 6 quarters, 5 dimes, 5 pennies, and some nickels, the ratio of nickels to dimes is 6:5.
To find the ratio of nickels to dimes, we need to determine the number of nickels and dimes in the pot. We know that there are 6 quarters, 5 dimes, and 5 pennies in the pot, which is a total of 16 coins. Therefore, the number of nickels should be the difference between the total number of coins and the sum of quarters, dimes, and pennies.
The total value of the coins in the pot is $2.35. Since 6 quarters are worth $1.50, 5 dimes are worth $0.50, and 5 pennies are worth $0.05, the remaining value should come from the nickels.
Thus, the value of the nickels is $2.35 - $1.50 - $0.50 - $0.05 = $0.30. Since each nickel is worth $0.05, the number of nickels is $0.30 ÷ $0.05 = 6.
The ratio of nickels to dimes is therefore 6:5, which means that for every 6 nickels, there are 5 dimes.
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Renee wants to put a fence around her Square Garden that has an area of 6500 square feet determine the perimeter of the garden to the nearest tenth of a foot
Answer:
299.3
Step-by-step explanation:
The nine squares of a 3-by-3 chessboard are to be colored red and blue. The chessboard is free to rotate but cannot be flipped over. Determine the generating function for the number of nonequivalent colorings and the total number of nonequivalent colorings.
Answer:
[tex]a_n = 2^{\frac{n^2-1}{4} + 1} + \frac{2^{n^2} - \, 2^{\frac{n^2-1}{4} + 1}}{4}[/tex]
For n = 3, there are 134 possibilities
Step-by-step explanation:
First, lets calculate the generating function.
For each square we have 2 possibilities: red and blue. The Possibilities between n² squares multiply one with each other, giving you a total of 2^n² possibilities to fill the chessboard with the colors blue or red.
However, rotations are to be considered, then we should divide the result by 4, because there are 4 ways to flip the chessboard (including not moving it), that means that each configuration is equivalent to three other ones, so we are counting each configuration 4 times, with the exception of configurations that doesnt change with rotations.
A chessboard that doesnt change with rotations should have, in each position different from the center, the same colors than the other three positions it could be rotated into. As a result, we can define a symmetric by rotations chessboard with only (n²-1)/4 + 1 squares (the quarter part of the total of squares excluding the center plus the center).
We cocnlude that the total of configurations of symmetrical boards is [tex] 2^{\frac{n^2-1}{4} + 1} [/tex]
Since we have to divide by 4 the rest of configurations (because we are counted 4 times each one considering rotations), then the total number of configutations is
[tex]a_n = 2^{\frac{n^2-1}{4} + 1} + \frac{2^{n^2} - \, 2^{\frac{n^2-1}{4} + 1}}{4}[/tex]
If n = 3, then the total amount of possibilities are
[tex]a_3 = 2^{\frac{3^2-1}{4} + 1} + \frac{2^{3^2} - \, 2^{\frac{3^2-1}{4} + 1}}{4} = 134[/tex]
Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x- and y-coordinates of P, Q, and R are to be integers that satisfy the inequalities -4 <= x <= 5 and 6<= y<= 16. How many different triangles with these properties could be constructed?
(A) 110
(B) 1,100
(C) 9,900
(D) 10,000
(E) 12,100
Answer:
(C) 9900
Step-by-step explanation:
The right triangle which right angle is at P and PR is parallel to the x-axis can have 4 sets (A,B,C,D as illustrated) of format depends on which direction is the right angle located.
each set have
(1+2+3+4+5+6+7+8+9) x (1+2+3+4+5+6+7+8+9+10) = 45 x 55 = 2475 right triangles
4 sets: 2475 x 4 = 9900
You're really good at investing and you have $1,500 in your investment account.You make 8.5% interest a year on your investment account!For a year you owe $1,600 on a credit card.You pay 19% interest a year on this credit card debt.Answer these questions:Some of the answers are incorrect. Try again...Here is a hint: How much are you making on your investment? Calculate: $1,500 * 8.5%. What do you pay in interest on your card? You do the math: $1,600 * 19%. Looks like a huge loss of money, right? Enter the loss as a negative value.How much money are you making on your investment in a year?$ How much money are you paying in interest in a year on your card?$ What's your total gain/loss that year?$ Enter a negative value for a loss.
Answer:
Step-by-step explanation:
To determine how much are you making on your investment, we will apply the simple interest formula. It is expressed as
I = PRT/100
Where
P = principal or amount invested,
R = interest rate
T = time
From the information given,
P = $1,500
T = 1 year
R = 8.5
I =( 1500×8.5×1)/100
I = $127.5
For a year you owe $1,600 on a credit card. You pay 19% interest a year on this credit card debt.
Interest paid = (1600×19×1)/100 = $304
The loss is 304 - 127.5 = - $176.5
Final answer:
You are making $127.50 on your investment in a year and paying $304 in interest on your credit card debt in a year, resulting in a total loss of $176.50.
Explanation:
First, let's calculate how much money you are making on your investment in a year. To do this, multiply your initial investment of $1,500 by the interest rate of 8.5%: $1,500 * 8.5% = $127.50. So, you are making $127.50 on your investment in a year.
Next, let's calculate how much money you are paying in interest on your credit card debt in a year. To do this, multiply your credit card debt of $1,600 by the interest rate of 19%: $1,600 * 19% = $304. So, you are paying $304 in interest on your card in a year.
To calculate your total gain/loss for the year, subtract the amount you are paying in interest from the amount you are making on your investment: $127.50 - $304 = -$176.50. Therefore, you have a total loss of $176.50 for the year. Remember to enter a negative value for a loss.
A rectangle has a side that is 16 feet and another side there's 1/2 that links a square has a perimeter of 48 feet how much greater is the area of the Square in the area of the rectangle
Answer:
Area of rectangle = 128 square feet
Step-by-step explanation:
Given:- A rectangle with side(a)=16 feet, side (b) = [tex]\frac{1}{2}[/tex] that links to a square.
perimeter (p) = 48 feet.
To find:- area of the square of the rectangle=?
Now,
[tex]Perimeter\ of\ square\ (p) = (2\times a)+(2\times b)[/tex]
[tex]48=(2\times 16)+(2\times b)[/tex]
[tex]48=32+2b[/tex]
[tex]2b=48-32[/tex]
[tex]2b=16[/tex]
[tex]b=\frac{16}{2}[/tex]
[tex]b=8 feet[/tex] -------(equation 1)
(8 is half of square of 4=16, [tex]4^{2}=16,\ \frac{16}{2} = 8[/tex])
Now, to find the area of square:-
Area of square (A) = Length [tex]\times[/tex] breadth
Area of square (A)= side a [tex]\times[/tex] side b
A= 16 [tex]\times[/tex] 8
[tex]\therefore[/tex]A = 128 square feet
Therefore Area of rectangle = 128 square feet
Natalie visits a grocery store to buy tomatoes. The cost of tomatoes is $26. She is remitted the bill and received $4 in change from the cashier. Write the equation to find how much she paid the cashier? Let m equal amount she paid
Answer:
m-4=26
Step-by-step explanation: I guess and feel like this is correct for some reason
What polynomial should be subtracted from 7x2−6x+5 to get the difference equal to x2−x.
Answer:
[tex]6x^2-5x+5[/tex] should be subtracted.
Step-by-step explanation:
we find the polynomial that is subtracted from 7x^2-6x+5 to get the difference equal to x^2-x
7x^2-6x+5-polynomial=x^2-x
To get the polynomial subtract x^2-x from the given polynomial
[tex]7x^2-6x+5 - (x^2-x)= 7x^2-6x+5-x^2+x[/tex]
[tex]6x^2-5x+5[/tex]
So [tex]6x^2-5x+5[/tex] should be subtracted.
Estimate 3.49x 7.508 by first rounding each number to the nearest whole number. Give your estimate as a whole number.
Answer:
24
Step-by-step explanation:
3.49 to the nearest whole number is 3
7.508 to the nearest whole number is 8
Multiplying both will yield 3 * 8 = 24
Although it might be mistaken that 3.49 could be approximated to 4, this is absolutely wrong. This is because we round up all values 5 and above after the decimal to 1 while we round down all values less than 5 after the decimal to 0.
Hence be it 3.4999, since it is less than 5, it is rounded as 3 to the nearest whole number digit
A group of friends decided to rent a house in Aspen, Colorado for a week of skiing. They each had to chip in $70 for the week’s lodging. If they had been able to convince three more people to go, the cost per person would have been reduced by $14. What was the rent for the week?
Answer:
70/5
Step-by-step explanation:
70/3 and then try 70/4 and then 70/5
The total rent for the week was $840, based on the given conditions of per person cost and the price decrease with additional participants.
Explanation:The subject of this question is Mathematics, and this is a problem ideally pitched at high school level. It involves constructing equations from the given information to solve the problem.
Let's begin by determining the amount of people who went on the trip initially. We'll call them 'n'. The cost per person on the trip was $70, so that the total cost of the trip is $70n.
Now, if they had persuaded three more people to go (n + 3), the cost per person would've dropped by $14 to $56 which, multiplied by the new total of attendees would still be equal to the total cost of the trip ($56(n + 3)).
As such, we build the equation $70n = $56(n + 3). Here's the breakdown and solving of the equation: $70n = $56n + $168.
Subtracting $56n from both sides gives $14n = $168. Dividing both sides by 14 finally gives n = 12.
To find the total cost of the rent, substitute n = 12 into the equation $70n, yielding $70(12) = $840.
So, the rent for the week was $840.
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There is a bag filled with marbles: 5 red, 8 blue, 4 yellow, and 3 green.
You want to draw a red then a blue marble. Do you have a better chance of drawing a red then a blue marble with or without replacing the first marble? Explain your answer.
need answer asap! if you could give me an explanation, that would be great! thank you and have a wonderful day!
Answer:
it depends
Step-by-step explanation:
If you draw a green one then you would do better without it but if you draw a red you would do better putting it back
Answer
no
Step-by-step explanation:
just because
At a hardware store a tool set normally cost $80 during the sale this week the tools that cost $12 less than usual what percentage of the usual price is a savings explain or show your reasoning
15 % of usual price is a savings
Solution:
Given that tool set normally cost $80
During the sale this week the tools that cost $12 less than usual
To find: what percentage of the usual price is a savings
From given information,
Usual price of tool set = $ 80
Given that tools that cost $12 less than usual which means she saved $ 12
Savings = $ 12
To find what percentage of the usual price is a savings, we can solve by framing a expression,
Let "x" be the required percentage
Then x % of percentage is equal to savings price
x % of usual price = savings price
x % of 80 = 12
[tex]\frac{x}{100} \times 80 = 12\\\\x = \frac{12 \times 100}{80}\\\\x = 15 \%[/tex]
Therefore 15 % of usual price is a savings
After the mill in a small town closed down in 1970, the population of that town started decreasing according to the law of exponential growth and decay. By 1990, the population had decreased to 143 thousand. By 2019, the population further decreased down to 98 thousand. Heat was the original population in 1970.
Answer:
[tex]P_o = \frac{143000}{e^{-20*0.01303024661}}=110193.69[/tex]
And we can round this to the nearest up integer and we got 110194.
Step-by-step explanation:
The natural growth and decay model is given by:
[tex]\frac{dP}{dt}=kP[/tex] (1)
Where P represent the population and t the time in years since 1970.
If we integrate both sides from equation (1) we got:
[tex] \int \frac{dP}{P} =\int kdt [/tex]
[tex]ln|P| =kt +c[/tex]
And if we apply exponentials on both sides we got:
[tex]P= e^{kt} e^k [/tex]
And we can assume [tex]e^k = P_o[/tex]
And we have this model:
[tex]P(t) = P_o e^{kt}[/tex]
And for this case we want to find [tex]P_o[/tex]
By 1990 we have t=20 years since 1970 and we have this equation:
[tex]143000 = P_o e^{20k}[/tex]
And we can solve for [tex]P_o[/tex] like this:
[tex]P_o = \frac{143000}{e^{20k}}[/tex] (1)
By 2019 we have 49 years since 1970 the equation is given by:
[tex]98000 = P_o e^{49k}[/tex] (2)
And replacing [tex]P_o[/tex] from equation (1) we got:
[tex]98000=\frac{143000}{e^{20k}} e^{49k} =143000 e^{29k}[/tex]
We can divide both sides by 143000 we got:
[tex]\frac{98000}{143000} =0.685 = e^{29k}[/tex]
And if we apply ln on both sides we got:
[tex]ln(0.685) = 29k[/tex]
And then k =-0.01303024661[/tex]
And replacing into equation (1) we got:
[tex]P_o = \frac{143000}{e^{-20*0.01303024661}}=110193.69[/tex]
And we can round this to the nearest up integer and we got 110194.
Evaluate the expression \dfrac{x^5}{x^2} x 2 x 5 start fraction, x, start superscript, 5, end superscript, divided by, x, squared, end fraction for x=2x=2x, equals, 2.
Answer:
8
Step-by-step explanation:
Fill in the variable value and do the arithmetic.
[tex]\dfrac{2^5}{2^2}=\dfrac{32}{4}=8[/tex]
___
Of course, the fraction can be simplified first:
[tex]\dfrac{x^5}{x^2}=x^{5-2}=x^3\\\\2^3=8[/tex]
To evaluate the expression, substitute x with 2, simplify the exponents, and perform the multiplication
Explanation:To evaluate the expression \dfrac{x^5}{x^2} \times 2 \times 5
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The length of the top of a computer desk is 2 1/4 feet longer than it's width. If it's width measures y feet, express its length as an algebraic expression in y
Answer:
Step-by-step explanation:
The top of the computer desk is rectangular in shape.
Let y represent the width of the rectangle.
The length of the top of the computer desk is 2 1/4 feet longer than its width. Converting 2 1/4 feet to improper fraction, it becomes 9/4 feet. Therefore, the algebraic expression of the length of the of the top of the computer desk in terms of y would be
Length = y + 9/4
Write the equation of the line that has a slope of 2 and passes through the point (-3,4).
A) y = 2x - 2
B) y = 2x + 2
C) y = 2x + 7
D) y = 2x + 10
Answer:
The answer to your question is letter D
Step-by-step explanation:
Data
slope = m = 2
Point (-3, 4)
Process
1.- Substitute the data in the line equation
y - y1 = m(x - x1)
y - 4 = 2 (x + 3)
2.- Expand
y - 4 = 2x + 6
3.- Solve for y and simplify
y = 2x + 6 + 4
y = 2x + 10
Answer:
y=2x+10
Step-by-step explanation:
Money in a particular savings account increases by about 6% after a year.How much money will be in the account after one year if the initial amount is $100 $50 $200 $125 x dollar?
Answer:
Part 1) [tex]\$106[/tex]
Part 2) [tex]\$53[/tex]
Part 3) [tex]\$212[/tex]
Part 4) [tex]\$132.50[/tex]
Part 5) [tex]\$1.06x[/tex]
Step-by-step explanation:
we have
Money in a particular savings account increases by about 6% after a year.
we know that
The simple interest formula is equal to
[tex]A=P(1+rt)[/tex]
where
A is the Final Investment Value
P is the Principal amount of money to be invested
r is the rate of interest
t is Number of Time Periods
Part 1) How much money will be in the account after one year if the initial amount is $100
in this problem we have
[tex]t=1\ year\\ P=\$100\\ A=?\\r=6\%=6/100=0.06[/tex]
substitute in the formula above
[tex]A=100(1+0.06*1)[/tex]
[tex]A=100(1.06)[/tex]
[tex]A=\$106[/tex]
Part 2) How much money will be in the account after one year if the initial amount is $50
in this problem we have
[tex]t=1\ year\\ P=\$50\\ A=?\\r=6\%=6/100=0.06[/tex]
substitute in the formula above
[tex]A=50(1+0.06*1)[/tex]
[tex]A=50(1.06)[/tex]
[tex]A=\$53[/tex]
Part 3) How much money will be in the account after one year if the initial amount is $200
in this problem we have
[tex]t=1\ year\\ P=\$200\\ A=?\\r=6\%=6/100=0.06[/tex]
substitute in the formula above
[tex]A=200(1+0.06*1)[/tex]
[tex]A=200(1.06)[/tex]
[tex]A=\$212[/tex]
Part 4) How much money will be in the account after one year if the initial amount is $125
in this problem we have
[tex]t=1\ year\\ P=\$125\\ A=?\\r=6\%=6/100=0.06[/tex]
substitute in the formula above
[tex]A=125(1+0.06*1)[/tex]
[tex]A=125(1.06)[/tex]
[tex]A=\$132.50[/tex]
Part 5) How much money will be in the account after one year if the initial amount is $x
in this problem we have
[tex]t=1\ year\\ P=\$x\\ A=?\\r=6\%=6/100=0.06[/tex]
substitute in the formula above
[tex]A=x(1+0.06*1)[/tex]
[tex]A=x(1.06)[/tex]
[tex]A=\$1.06x[/tex]
1. what is x? (picture 1 and 2)
2. What is the length of NO? (3rd picture)
3. If LB = 6 and LN = 2x+5, what is x? (fourth picture)
Answer:
Part 1) [tex]x=6[/tex]
Part 2) [tex]x=5,75[/tex]
Part 3) [tex]NO=80\ units[/tex]
Part 4) [tex]x=3,5[/tex]
Step-by-step explanation:
Part 1) Find the value of x
we know that
In a parallelogram opposites sides are congruent and parallel
In this problem
GH=FE
substitute the given values
[tex]2x+10=22[/tex]
solve for x
subtract 10 both sides
[tex]2x=22-10[/tex]
[tex]2x=12[/tex]
Divide by 2 both sides
[tex]x=6[/tex]
Part 2) Find the value of x
we know that
In a parallelogram opposites sides are congruent and parallel
In this problem
FG=EH
substitute the given values
[tex]4x+5=28[/tex]
solve for x
subtract 5 both sides
[tex]4x=28-5[/tex]
[tex]4x=23[/tex]
divide by 4 both sides
[tex]x=5,75[/tex]
Part 3) What is the length of NO?
step 1
Find the value of x
we know that
In a parallelogram opposites sides are congruent and parallel
In this problem
NO=ML
substitute the given values
[tex]4x+20=2x+50[/tex]
solve for x
Group terms
[tex]4x-2x=50-20[/tex]
[tex]2x=30[/tex]
Divide by 2 both sides
[tex]x=15[/tex]
step 2
Find the value of NO
we have that
[tex]NO=4x+20[/tex]
substitute the value of x
[tex]NO=4(15)+20=80\ units[/tex]
Part 4) we know that
The diagonals in a parallelogram bisect each other
so
LB=BN
LN=LB+BN ----> by addition length postulate
LN=2LB
substitute the given values
[tex]2x+5=2(6)[/tex]
solve for x
[tex]2x+5=12[/tex]
subtract 5 both sides
[tex]2x=12-5[/tex]
[tex]2x=7[/tex]
Divide by 2 both sides
[tex]x=3,5[/tex]
The second term in a geometric sequence is 81. The common ratio for the geometric sequence is 3. Use the common ratio or equation to find the 4th and 6th terms in the geometric sequence.
Show your work
Answer:4th: 729 6th: 6561
Step-by-step explanation: Sorry, I’m not 100% sure but I will try to help out: :)
So 2nd term is 81 and you want the 4th and 6th term.
Common ratio is 3
I approached it like this:
81x3=243 (3rd Term)
243x3=729 (4th Term)
729x3=2187 (5th Term)
2187x3=6561 (6th Term)
So if this isn’t the correct way the only other way I can think to approach this is
81+3=84(3rd Term)
84+3=87(4th Term)
87+3=90(5th Term)
90+3=93(6th Term)
Hope this helps
Final answer:
The 4th term is 729 and the 6th term is 6561 in the given geometric sequence with a common ratio of 3, starting with the second term of 81.
Explanation:
To find the 4th and 6th terms in a geometric sequence, we use the formula for the nth term of a geometric sequence Tn = ar^(n-1), where a is the first term, r is the common ratio, and n is the term number.
Given that the second term is 81 and the common ratio (r) is 3, we can find the first term by using the second term's formula: T2 = ar^(2-1) = ar = 81, so a = 81/r = 81/3 = 27.
Now, to find the 4th term (T4), we substitute the values into the formula: T4 = ar^(4-1) = 27 * 3^(3) = 27 * 27 = 729.
Similarly, to find the 6th term (T6), we use the formula again: T6 = ar^(6-1) = 27 * 3^(5) = 27 * 243 = 6561.
In conclusion, the 4th term is 729 and the 6th term is 6561 in this geometric sequence.
John weighs three times as much as Karen. Two times John's weight plus Karen's weight is 875 pounds. How much does John weigh? How much does Karen weigh?
Answer:
John- 375
Karen- 125
Step-by-step explanation:
Answer 1...
j =3k
2j + k = 875
substituting the first eqn into the 2nd
2(3k) + k = 875
6k+ k =875
7k = 875
k =875/7 =125
thus j = 3(125) =375
Answer:john weighs 375 pounds.
Karen weighs 125 pounds
Step-by-step explanation:
Let x represent the weight of John.
Let y represent the weight of Karen.
John weighs three times as much as Karen. This means that
x = 3y
Two times John's weight plus Karen's weight is 875 pounds. This means that
2x + y = 875 - - - - - - - -1
Substituting x = 3y into equation 1, it becomes
2 × 3y + y = 875
6y + y = 875
7y = 875
y = 875/7 = 125
Substituting y = 125 into x = 3y. It becomes
x = 3 × 125 = 375
How many possible 4-digit combinations are there with the numbers 2, 3, 4, 5, 6, 7, 8, and 9 if none of the numbers appear more than once (i.E. 2343, 2333, 2323, etc.)?
Answer:
[tex]1680[/tex]
Step-by-step explanation:
we need a 4-digit number from the numbers [tex]2,3,4,5,6,7,8\ and\ 9[/tex] (without repetition ).
possible number at thousand place [tex]=8[/tex]
Possible numbers at hundred place[tex]=8-1=7[/tex]
Possible numbers at [tex]10^{th}[/tex] place [tex]=7-1=6[/tex]
possible number at unit place [tex]=6-1=5[/tex]
So total possible numbers
[tex]=8\times7\times6\times5\\=1680[/tex]
Other method :
We are taking [tex]4[/tex] numbers out of [tex]8[/tex] and here order matters so we will use permutation.
Total possible numbers [tex]=^8P_{4}[/tex]
[tex]\frac{8!}{(8-4)!}\\ =\frac{8!}{4!}\\ =8\times7\times6\times5\\=1680[/tex]
What is the Common difference in the sequence 10,20,30,40,50...?
Answer:
10
Step-by-step explanation:
divide the last number by the previous number
Answer:
10
Step-by-step explanation:
If you calculate it correctly, every number in front of that number it 10 above.
Water evaporates from a pond at aa rate of 0.005 inches per hour. What is the change in the water level of the pond after 24 hours? Express your answer as a rational number
Answer:
Water level after 24 hours will be [tex]\frac{3}{25}\ inches[/tex].
Step-by-step explanation:
Given:
Evaporate rate of water= 0.005 inches per hour.
We need to find the level of water after 24 hours.
now we know the in 1 hour the water evaporates from a pool at a rate of 0.005 inches.
So To find the water level after 24 hrs we will multiply the evaporation rate with total number of hours which is 24 and the divide by 1 hour we get.
Framing in equation form we get;
water level after 24 hrs = [tex]\frac{24 \times 0.005}1 = 0.12\ inches[/tex]
Now to convert the number in rational form we have to multiply and divide the number by 100 we get;
[tex]\frac{0.12\times 100}{100} = \frac{12}{100}=\frac{3}{25}\ inches[/tex]
Hence Water level after 24 hours will be [tex]\frac{3}{25}\ inches[/tex].
A certain company assigns employees to offices in such a way that some of the offices can be empty and more than one employee can be assigned to an office. In how many ways can the company assign 3 employees to 2 different offices?A. 5B. 6C. 7D. 8E. 9
Answer: The answer is 6
Step-by-step explanation: this is a combination because it is without repetition.
3!/(3-2)!
(3x2x1)/1!
6/1 = 6
So the answer is 6 different ways
Diane loves coasters that dip into tunnels during the ride.Her favorite coaster is modeled by h(t)=2t +23t-59t+24. Using rational route theorem, what are the possible rational zeros for the function
Answer:
The possible rational zeros for the function are
±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, ±1/2, ±3/2
Step-by-step explanation:
I believe that there is an error in the function with the exponents, it must be:
[tex]h(t) = 2t^{3} + 23t^{2}+59t+24[/tex]
If this is the function that you need, then we must use the rational zero theorem. It says that if a polynomial function, written in descending order of the exponents, has integer coefficients, then any rational zero must be of the form ± p/ q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Thus
In this case the constant term is 24 and then
p = ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
The factor of the leading coefficient is 2, thus
q = ±1, ±2
The possible rational zeros for the function are
±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, ±1/2, ±3/2
A girl is now one-fourth as old as her father, and in seven years, she will be one-half as old as her father was twelve years ago. What are her and her father's present ages?A. father's age = 20; daughter's age = 5B. father's age = 52; daughter's age = 13C. father's age = 76; daughter's age = 19
Answer:
Option B - father's age = 52; daughter's age = 13
Step-by-step explanation:
Given : A girl is now one-fourth as old as her father, and in seven years, she will be one-half as old as her father was twelve years ago.
To find : What are her and her father's present ages?
Solution :
Let the father's present age is 'x'.
A girl is now one-fourth as old as her father.
i.e. Girl age is [tex]\frac{x}{4}[/tex]
In seven years, she will be one-half as old as her father was twelve years ago.
i.e. [tex]\frac{x}{4}+7=\frac{1}{2}(x-12)[/tex]
[tex]\frac{x}{4}+7=\frac{x}{2}-6[/tex]
[tex]\frac{x}{4}-\frac{x}{2}=-6-7[/tex]
[tex]\frac{x-2x}{4}=-13[/tex]
[tex]-x=-52[/tex]
[tex]x=52[/tex]
The father's age is 52 years.
The daughter's age is [tex]\frac{52}{4}=13[/tex]
Therefore, option B is correct.
A gun with a muzzle velocity of 1500 feet per second is fired at an angle of 6 degrees with the horizontal. Find the vertical and horizontal components of the velocity to the nearest whole number.
Answer:
The vertical component of velocity is 156 feet per second
The horizontal component of velocity 1491 feet per second .
Step-by-step explanation:
Given as :
The velocity of gun = v = 1500 feet per sec
The angle made by gun with horizontal = Ф = 6°
Let The vertical component of velocity = [tex]v__y[/tex]
Let The horizontal component of velocity = [tex]v__x[/tex]
Now, According to question
The vertical component of velocity = v sin Ф
i.e [tex]v__y[/tex] = v sin Ф
Or , [tex]v__y[/tex] = 1500 ft/sec × sin 6°
Or , [tex]v__y[/tex] = 1500 ft/sec × 0.104
∴ [tex]v__y[/tex] = 156 feet per second
So, The vertical component of velocity = [tex]v__y[/tex] = 156 feet per second
Now, Again
The horizontal component of velocity = v cos Ф
i.e [tex]v__x[/tex] = v cos Ф
Or , [tex]v__x[/tex] = 1500 ft/sec × cos 6°
Or , [tex]v__x[/tex] = 1500 ft/sec × 0.994
∴ [tex]v__x[/tex] = 1491 feet per second
So, The horizontal component of velocity = [tex]v__y[/tex] = 1491 feet per second
Hence,The vertical component of velocity is 156 feet per second
And The horizontal component of velocity 1491 feet per second . Answer
Final answer:
The horizontal component of the bullet's velocity is 1492 feet per second, and the vertical component is 157 feet per second, rounded to the nearest whole number, when fired from a gun at an angle of 6 degrees with a muzzle velocity of 1500 feet per second.
Explanation:
The student is asking to find the vertical and horizontal components of a bullet's velocity when fired from a gun at a specific angle. To solve this, we use trigonometric functions, specifically sine and cosine, since the bullet's velocity makes an angle with the horizontal axis. Given a muzzle velocity of 1500 feet per second and an angle of 6 degrees with the horizontal, the horizontal component (Vx) is V * cos(θ) and the vertical component (Vy) is V * sin(θ).
Calculating the horizontal component: Vx = 1500 * cos(6 degrees) = 1500 * 0.99452 ≈ 1492 feet per second.
Calculating the vertical component: Vy = 1500 * sin(6 degrees) = 1500 * 0.10453 ≈ 157 feet per second.
We round these to the nearest whole number as per the question's requirement, so the horizontal component is 1492 feet per second and the vertical component is 157 feet per second.