point P is the image of p(-2,-2) translated by 1 unit to the left and 3 units now

Answers

Answer 1

To find the image of point P(-2,-2) translated 1 unit to the left and 3 units down, we subtract 1 from the x-coordinate and 3 from the y-coordinate to get coordinates of P' as: (-3, -5).

How to Find the Coordinates in Translation?

To translate a point to the left, we subtract from its x-coordinate, and to translate it down, we subtract from its y-coordinate.

Therefore, to translate P(-2, -2) 1 unit to the left and 3 units down, we subtract 1 from the x-coordinate and 3 from the y-coordinate to get P'(-3, -5) as the image of P after translation.

Therefore, the coordinates of P' are (-3, -5).

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Complete Question:

Point P' is the image of P(-2,-2) translated by 1 unit to the left and 3 units down. What are the coordinates of P'?


Related Questions

Patricia bought 4 apples and 9 bananas for $12. 70. Jose bought 8 apples and 11 bananas for $17. 70 at the same grocery store. What's the price of one apple?

Answers

The price of one apple is $0.70, obtained by solving the system of equations 4x + 9y = 12.70 and 8x + 11y = 17.70 using elimination.

How much would Patricia pay for each apples?

Let's use a system of equations caculation the problem.

Let x be the price of one apple and y be the price of one banana.

From the first sentence, we know that:

4x + 9y = 12.70

From the second sentence, we know that:

8x + 11y = 17.70

Now we can solve for x by using either substitution or elimination.

Let's use elimination.

We can multiply the first equation by 11 and the second equation by -9, then add them together:

44x + 99y = 139.70

-72x - 99y = -159.30

-28x = -19.60

Dividing both sides by -28, we get:

x = 0.70

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What is the area of EFG

Answers

Answer:

A = 28 units²

Step-by-step explanation:

the area (A) of a triangle is calculated as

A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the perpendicular height )

here b = FG = 8 and h = FE = 7 , then

A = [tex]\frac{1}{2}[/tex] × 8 × 7 = 4 × 7 = 28 units²

PLEASE HELP!! Find the area of the figure.

Answers

The area of the trapezoid in this problem is given as follows:

15 square feet.

How to obtain the height of the trapezoid?

The area of a trapezoid is given by half the multiplication of the height by the sum of the bases, hence:

A = 0.5 x h x (b1 + b2).

The dimensions for this problem are given as follows:

h = 3 ft, b1 = 4 ft and b2 = 6 ft.

Hence the area is given as follows:

A = 0.5 x 3 x (4 + 6)

A = 1.5 x 10

A = 15 square feet.

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Use the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluated without a calculator. In(8x2 – 72x + 112) Enter the

Answers

The fully expanded expression using logarithm properties is:

[tex]In(8x^2 - 72x + 112) = 2.079 + In(x - 2) + In(x –-7)[/tex]

How to expand an expression?

To expand the given expression[tex]In(8x^2 - 72x + 112)[/tex], we can use the following logarithmic properties:

Product Rule: [tex]logb (xy) = logb x + logb y[/tex]

Quotient Rule: [tex]logb (x/y) = logb x - logb y[/tex]

Power Rule:[tex]logb (x^a) = a logb x[/tex]

We can first factor out a common factor of 8 from the expression inside the logarithm:

[tex]In(8x^2 - 72x + 112) = In[8(x^2 - 9x + 14)][/tex]

Using the distributive property, we can expand the expression inside the logarithm:

[tex]In[8(x^2 - 9x + 14)] = In(8) + In(x^2 - 9x + 14)[/tex]

Now, we need to expand the second logarithm. We notice that the expression inside the logarithm can be factored as follows:

[tex]x^2 - 9x + 14 = (x - 2)(x - 7)[/tex]

Using the product rule, we can write:

[tex]In(x^2 - 9x + 14) = In[(x - 2)(x - 7)][/tex]

[tex]= In(x - 2) + In(x - 7)[/tex]

Putting all the pieces together, we get:

[tex]In(8x^2 - 72x + 112) = In(8) + In(x^2 - 9x + 14)[/tex]

[tex]= In(8) + In(x -2) + In(x - 7)[/tex]

Finally, we can simplify the numerical expression In(8) by using the fact that ln(e) = 1:

[tex]In(8) = ln(8)/ln(e) = 2.079/1 = 2.079[/tex]

So the fully expanded expression using logarithm properties is:

[tex]In(8x^2 - 72x + 112) = 2.079 + In(x - 2) + In(x -7)[/tex]

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The arrow on this spinner is equally likely to land on each section. the arrow is spun 72 times. how many times do you expect the arrow to land on 4?

Answers

we know that the spinner has an equal chance of landing on each section. Since there are a total of six sections on the spinner, we can assume that the probability of the arrow landing on any one section is 1/6 or approximately 0.1667.

Now, if the arrow is spun 72 times, we can use this probability to calculate the expected number of times the arrow will land on 4. To do this, we simply multiply the probability by the number of spins, as follows:

Expected number of times arrow lands on 4 = Probability of arrow landing on 4 x Number of spins
Expected number of times arrow lands on 4 = 0.1667 x 72
Expected number of times arrow lands on 4 = 12

So, we can expect the arrow to land on 4 approximately 12 times out of 72 spins. Of course, this is just an expected value, and the actual number of times the arrow lands on 4 may vary from this value due to random chance.

In summary, if we assume that the arrow on the spinner is equally likely to land on each section, and it is spun 72 times, we can expect the arrow to land on 4 approximately 12 times based on probability calculations.

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The line plot shows the ages of participants in the middle school play
Determine the appropriate measures of center and variation

Answers

The appropriate measures of center and variation are 13 and 4.

Measures of Center:

The appropriate measure of center for this data set is the median since there is no clear outlier present in the data. Hence, the value of median here is 13

Measures of Variation:

The appropriate measure of variation for this dataset is the range, which is the difference between the largest and smallest value in the dataset, Hence the value of range is 4

Since the data is small and there is no clear outlier present, the median is the appropriate measure of center. The range, which is the difference between the largest and smallest value in the dataset, is the appropriate measure of variation

Hence, the appropriate measures of center and variation are 13 and 4.

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WHY CANT YOU JUST GIVE AN ANSWER WITHOUT MAKING THE PERSON PAY I JUST WANT A EXPLANAITION FOR A QUESTION STILL YOUR MAKING ME PAY ME JUST FOR A ANSWER AND A SIMPLE EXPLANAITION YOUR ADDS ALWAYS FREEZE SO NOW K HAVE TO PAY? OH MY GOD EVERY OTHER WEBSITE DOES THE SAME THING WHY DO YOU DO THAT WITH THE REST IM JUST A GUEST oop sorry caps lock.

Answers

steal it

Step-by-step explanation:

Find a formula for the sum of n terms. Use the formula to find the limit as n = [infinity].
lim ∑ ( 6 + i/n) (2/n)

Answers

To find a formula for the sum of n terms, we need to first write out the first few terms of the series and look for a pattern:

n=1: (6+1/1) (2/1) = 14
n=2: (6+1/2) (2/2) + (6+2/2) (2/2) = 16
n=3: (6+1/3) (2/3) + (6+2/3) (2/3) + (6+3/3) (2/3) = 17 1/3
n=4: (6+1/4) (2/4) + (6+2/4) (2/4) + (6+3/4) (2/4) + (6+4/4) (2/4) = 18

From this, we can see that the nth term is given by (6+i/n) (2/n). To find the sum of n terms, we simply add up all of the terms from i=1 to i=n:

∑ (6+i/n) (2/n) = (2/n) ∑ (6+i/n)

Using the formula for the sum of an arithmetic series, we get:

∑ (6+i/n) = n/2 (6 + (6+n)/n)

Substituting this back into our expression for the sum of n terms, we get:

∑ (6+i/n) (2/n) = (2/n) * (n/2) * (6 + (6+n)/n) = 6 + (6+n)/n

Taking the limit as n approaches infinity, we get:

lim (6 + (6+n)/n) = 6 + lim ((6+n)/n) = 6 + 1 = 7

Therefore, the limit of the given series as n approaches infinity is 7.

To find the formula for the sum of n terms, we will use the concept of Riemann sums. Given the expression you provided, it appears that you are trying to compute the limit of the Riemann sum as n approaches infinity, which will give you the integral of the function.

Expression: lim (n→∞) ∑ (6 + i/n) (2/n)

First, let's rewrite the Riemann sum in integral form:

∫(6 + x)dx

Now we need to find the integral of the function and evaluate it over a specific interval. However, you haven't provided the interval, so I'll assume it is [a, b].

∫(6 + x)dx evaluated from a to b will give us the formula for the sum of n terms:

F(x) = 6x + (1/2)x^2

Now, evaluate F(x) over the interval [a, b]:

F(b) - F(a) = [6b + (1/2)b^2] - [6a + (1/2)a^2]

This is the formula for the sum of n terms. To find the limit as n approaches infinity, you will need to provide the specific interval [a, b]. Otherwise, the limit cannot be determined without further information.

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Two different furniture manufacturers produce chairs. let x represent the number of chairs produced daily at plant x, and let y represent the number of chairs produced daily at plat y

Answers

Sure, happy to help! So, we have two furniture manufacturers producing chairs, and we'll call them Plant X and Plant Y. Let x represent the number of chairs produced daily at Plant X, and let y represent the number of chairs produced daily at Plant Y.

Now, we don't know what the actual numbers are, but we can use these variables to talk about them in a general way. For example, we could say that Plant X produces 100 chairs per day (so x = 100), and Plant Y produces 200 chairs per day (so y = 200).

Does that make sense? Let me know if you have any other questions!

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Help with the problem in photo

Answers

The measure of the unknown arc is 76°

What is arc angle relationship?

An arc is a smooth curve joining two endpoints. It can also be defined as a portion of a circle.

Arc angle relationship states that If two chords intersect inside a circle, then the measure of each angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Therefore angle 30° = 1/2 ( x-61)

30 = 1/2( x -61)

15 = x -61

x = 15 + 61

x = 76°

therefore the measure of the unknown arc is 76°

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A backyard is 40. 5 feet long and 25 feet wide. in order to install a pool, the yard needs to be reduced by a scale of one-third. what is the area of the reduced yard? feet2.

Answers

If A backyard is 40. 5 feet long and 25 feet wide then, the area of the reduced yard is approximately 450.09 ft².

The area of the original backyard is:

40.5 ft x 25 ft = 1012.5 ft²

To reduce the yard by a scale of one-third, we need to multiply the length and width by 2/3:

40.5 ft x 2/3 = 27 ft

25 ft x 2/3 = 16.67 ft (rounded to two decimal places)

The area of the reduced yard is:

27 ft x 16.67 ft = 450.09 ft² (rounded to two decimal places)

Therefore, the area of the reduced yard is approximately 450.09 ft².

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equation of a line with slope m=−2/5 that contains the point (10,−5).

Answers

The equation of a line with slope m = -2/5 that contains the point (10,-5) can be found using the point-slope form of the equation of a line:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope.

Substituting the given values, we get:

y - (-5) = (-2/5)(x - 10)

Simplifying the right side, we get:

y + 5 = (-2/5)x + 4

Subtracting 5 from both sides, we get:

y = (-2/5)x - 1

Therefore, the equation of the line with slope m = -2/5 that contains the point (10,-5) is y = (-2/5)x - 1.

Answer:

y = (-2/5)x+b

Step-by-step explanation:

First plug these into the y=mx+b equation:

-5 = (-2/5)(10)+b.

Then solve for b:

-5 = -4+b

Add 4 to both sides:

-1 =b.

Therefore, the equation of the line is y = (-2/5)x+b. You can also double check this by plugging 10 into the equation we just obtained.

help please
Use substitution to find the indefinite integral. x² - 2x 3 X - S dx x4 - 4x + 4 - X x - 2x 4 - 4x + 4 dx=

Answers

To solve this problem, we can use substitution. Let's set u = x-2. Then, du/dx = 1 and dx = du. Using this substitution, we can rewrite the integral as:

∫ (u+2)² - 2(u+2) 3 (u+2) - Su du
Expanding the terms inside the integral, we get:
∫ (u² + 4u + 4) - 2(u+2)³ (u+2) - Su du
Simplifying, we get:


∫ u⁴ - 4u³ + 4u² - u³ + 6u² - 12u - u² + 6u - 9 du
Combining like terms, we get:
∫ u⁴ - 5u³ + 9u² - 6u - 9 du
Now, we can integrate each term separately using the power rule of integration:
∫ u⁴ - 5u³ + 9u² - 6u - 9 du = (1/5)u⁵ - (5/4)u⁴ + (9/3)u³ - 3u² - 9u + C
Substituting back u = x-2, we get:
(1/5)(x-2)⁵ - (5/4)(x-2)⁴ + (3)x³ - 3(x-2)² - 9(x-2) + C
Therefore, the indefinite integral of x² - 2x 3 X - S dx x⁴ - 4x + 4 - X x - 2x⁴ - 4x + 4 dx is (1/5)(x-2)⁵ - (5/4)(x-2)⁴ + (3)x³ - 3(x-2)² - 9(x-2) + C.
Hi! I'd be happy to help you with your integration problem. To find the indefinite integral using substitution, let's first rewrite the given integral:
∫(x² - 2x) / (x⁴ - 4x² + 4) dx



Now, let's perform substitution:
Let u = x² - 2x
Then, du/dx = 2x - 2
And also let v = x⁴ - 4x² + 4
Then, dv/dx = 4x³ - 8x
We need to find du in terms of dx, so:
du = (2x - 2) dx
Now, we can rewrite the integral in terms of u and v:
∫(u) / (v) (du / (2x - 2))
Now we can integrate:
(1/2) ∫(u) / (v) du



Unfortunately, this integral does not have a straightforward elementary antiderivative. However, you can use numerical integration methods or special functions to approximate the indefinite integral if necessary.

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Calculate the discount period for the bank to wait to receive its money. (Use table value):
Date of note Length of note Date note discounted Discount period
April 3 82 days May 10 days

Answers

The discount period for the bank to wait to receive its money is 37 days, and the discount rate is 3.5%.

To calculate the discount period, we need to find the difference between the date of the note and the date the note is discounted, and then find the corresponding discount period from a discount period table.

Date of note: April 3

Length of note: 82 days

Date note discounted: May 10

To find the number of days between April 3 and May 10, we can use a calendar or a date calculator, which gives us 37 days.

Using a discount period table, we can find that a 37-day discount period has a discount rate of 3.5%.

Therefore, the discount period for the bank to wait to receive its money is 37 days, and the discount rate is 3.5%.

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What adds to +10 and multiplys to 290

Answers

+10 * 290 = 2,900
The answer is 2,900

Find dx/dt at x = -5 if y = -5x^2 + 2 and dy/dt = - 4.
dx/dt = ?

Answers

x = -5, dx/dt is equal to -2/25.

To find dx/dt, we need to use the chain rule of differentiation.

We know that dy/dt = -4 and we have the equation y = -5x^2 + 2.

Taking the derivative of both sides with respect to t, we get:

dy/dt = d/dt (-5x^2 + 2)

Using the chain rule, we can write this as:

dy/dt = (-10x) (dx/dt)

Now, we can plug in x = -5 and dy/dt = -4:

-4 = (-10(-5)) (dx/dt)

Simplifying, we get:

-4 = 50 (dx/dt)

Dividing both sides by 50, we get:

dx/dt = -4/50

Simplifying further, we get:

dx/dt = -2/25

Therefore, at x = -5, dx/dt is equal to -2/25.

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Find f'(-2) for f(x) = ln((x^4 + 5)^2). Answer as an exact fraction or round to at least 2 decimal places.

Answers

Using the chain rule, we have: f'(x) = 2ln(x^4 + 5) * 2(x^4 + 5)^1 * 4x^3
f'(x) = 16x^3 * ln(x^4 + 5) * (x^4 + 5)

To find f'(-2), we plug in -2 for x:

f'(-2) = 16(-2)^3 * ln((-2)^4 + 5) * ((-2)^4 + 5)
f'(-2) = -128 * ln(21) * 21
f'(-2) ≈ -599.92 (rounded to 2 decimal places)

Therefore, f'(-2) is approximately -599.92.
To find f'(-2) for the function f(x) = ln((x^4 + 5)^2), we will first find the derivative of the function, and then evaluate it at x = -2.

1. Differentiate the function using the chain rule:
f'(x) = (d/dx) ln((x^4 + 5)^2) = (1/((x^4 + 5)^2)) * (d/dx) ((x^4 + 5)^2)

2. Differentiate the inner function:
(d/dx) ((x^4 + 5)^2) = 2(x^4 + 5) * (d/dx) (x^4 + 5) = 2(x^4 + 5) * (4x^3)

3. Combine the derivatives:
f'(x) = (1/((x^4 + 5)^2)) * (2(x^4 + 5) * (4x^3)) = (8x^3(x^4 + 5))/((x^4 + 5)^2)

4. Evaluate the derivative at x = -2:
f'(-2) = (8(-2)^3((-2)^4 + 5))/((-2)^4 + 5)^2 = (-128(21))/(21^2) = -128/21

So, f'(-2) is -128/21 as an exact fraction.

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A museum groundskeeper is creating a simicircular stauary garden with a diameter of 38 feet there will be a fence around the garden the fencing cost $9.25 per linear foot . About how much will the fencing cost although? Round to the nearest hundredth use 3.14 for n the fencing will cost about $

Answers

The amount for the fencing cost  is $903. 36

How to determine the value

From the information given, we have that the shape of the garden is  semi -circle.

Now, the formula that is used for calculating the circumference of a semicircle is expressed as;

C = πr + 2r

Given that the parameters of the equation are;

C is the circumference of the semicircler is the radius of the semicircle

From the information given,

Substitute the values, we have;

Circumference = 3.14(19) + 2(19)

expand the bracket

Circumference = 59. 66 + 38

Add the values

Circumference = 97. 66 feet

Then,

if 1 feet = $9.25

Then, 97. 66 feet = x

x = $903. 36

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Would anyone be willing to help me out with a few math questions? I'm up late and could really use the help!

Answers

State your questions and I’ll be happy to help

which fraction is equivalent to 0.48 in simplest form?

[A] 12/25
[B] 12/50
[c] 24/50
[D] 48/100

Answers

Answer:

0.48 = 48/100

48/100 ÷ 4/4 = 12/25

0.48 = 12/25 =A

2


A rhombus has a perimeter of 136 inches and one diagonal of 60 in.


What is the length of the other diagonal?


Find the area of the rhombus. .


Diagonal =


in


Area =


in2

Answers

The area of the rhombus is 1140 square inches.

Let the side length of the rhombus be "a" and let the length of the other diagonal be "d".

Since a rhombus has all sides congruent, the perimeter is given by:

4a = 136

Simplifying, we get:

a = 34

We can use the formula for the area of a rhombus:

Area = (diagonal 1 x diagonal 2)/2

Substituting the given values:

Area = (60 x d)/2

Area = 30d

Now we can substitute the value of "a" in terms of "d" into the formula for the length of the diagonal:

d = √(a² + b²)

d = √(34² + b²)

d = √(1156 + b²)

We also know that the perimeter of the rhombus is given by:

4a = 136

Substituting the value of "a" we found earlier:

4(34) = 136

So the length of the other diagonal can be found by subtracting the length of the given diagonal from the perimeter, and dividing by 2:

d = (136 - 60)/2 = 38

Therefore, the length of the other diagonal is 38 inches.

To find the area, we can substitute the value we found for "d" into the formula we derived earlier:

Area = 30d = 30(38) = 1140

So the area of the rhombus is 1140 square inches.

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At Olivia's Hats, 90% of the 80 hats are baseball caps. How many baseball caps are there?

Answers

pretty sure it’s 72

A recipe requires only blueberries and strawberries. This list shows the amounts required for 1/4 of the whole recipe:
1/2 cup blueberries
2/5 cup of strawberries

What is the number of cups of blueberries and the number of cups of strawberries required for the whole recipe?
a) 1/8 cup of blueberries and 1/10 cup of strawberries
b) 1/8 cup of blueberries and 1 3/5 cups of strawberries
c) 2 cups of blueberries and 1/10 cup of strawberries
d) 2 cups of blueberries and 1 3/5 cups of strawberries

Answers

A

Either divide each by one fourth or multiply each by 0.25. Then turn the answer to a fraction.

Can someone please answer the question below (Level: Year 8 (7th Grade) ) about algebra equations?
Thanks ^^

Answers

A) x=7.5
B) x=1

Explanation: solve for x by using inverse operation

Work:

The area of a square with side length s is s2. Meg crocheted a baby blanket for her new cousin. The blanket is a square with 30-inch sides. What is the area of the baby blanket? Write your answer as a whole number or decimal

Answers

The area of the baby blanket with side length of 30 inches is equal to 900 square inches.

Let 'A' represents the area of the square.

And s represents the side length of the square.

The area of a square is given by the formula

A = s^2.

For Meg's baby blanket,

The side length of the baby blanket is equal to 30 inches,

Substitute the values in the area formula we get,

A = s^2

⇒ A = 30^2

⇒ A = 900 square inches

Therefore, the area of Meg's baby blanket is equal to 900 square inches.

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Kera took out a 24-month bank loan of $13,000 at an interest rate of 5. 95%. She budgets to pay


$450 per month towards the loan. Write an equation that represents how much total interest


Kera will pay towards the remaining balance of the loan at the end of each year. Let m equal the


number of months paid and r equal the interest charged on the remaining balance

Answers

The equation that represents how much total interest Kera will pay towards the remaining balance of the loan at the end of each year is Total Interest Paid = (Remaining Balance) x (Annual Interest Rate) = $422.03.

The equation that represents how much total interest Kera will pay towards the remaining balance of the loan at the end of each year is:

Total Interest Paid = (Remaining Balance) x (Annual Interest Rate)

To calculate the remaining balance after m months, we can use the formula for the present value of an annuity:

Remaining Balance = (Payment per Month) x ((1 - (1 + r)^(-n)) / r)

where r is the monthly interest rate (0.0595 / 12 = 0.004958), n is the total number of months (24), and m is the number of months paid (12, 24, etc.).

Plugging in the given values, we get:

Remaining Balance = 450 x ((1 - (1 + 0.004958)^(-12)) / 0.004958) = $6,752.45

To calculate the annual interest rate, we can use the formula:

Annual Interest Rate = (1 + r)^12 - 1

Plugging in the monthly interest rate, we get:

Annual Interest Rate = (1 + 0.004958)^12 - 1 = 0.0625

Therefore, the equation that represents how much total interest Kera will pay towards the remaining balance of the loan at the end of each year is:

Total Interest Paid = $6,752.45 x 0.0625 = $422.03 (rounded to the nearest cent)

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The area of a rooftop can be
expressed as (x + 9)2. The rooftop
is a rectangle with side lengths
that are factors of the expression
describing its area. Which expression
describes the length of one side of
the rooftop?

Answers

The expression that describes the length of one side of the rooftop is therefore: x - 9.

What is expression?

In mathematics, an expression is a combination of one or more variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. An expression can be as simple as a single variable or constant, or it can be a more complex combination of variables and operations.

Here,

The expression for the area of the rooftop is (x + 9)², where x is a variable representing the length of one side of the rectangle. To find the factors of this expression, we can expand it using the identity (a+b)² = a² + 2ab + b².

Expanding (x + 9)², we get:

(x + 9)² = x² + 18x + 81

Now, we need to find the factors of this expression that are also factors of the length of the sides of the rectangle. Since the sides of the rectangle must have a common factor of x, we can factor out x from the expression:

x² + 18x + 81 = x(x + 18) + 81

The factors of (x + 9)² are x(x + 18) + 81, (x + 9)(x + 9), (x - 9)(x - 9), and -(x + 9)(x + 9).

Since we are looking for factors that represent the length of one side of the rooftop, we can eliminate the negative factor and the factor (x + 9)(x + 9), since the sides of a rectangle must be positive.

That leaves us with x(x + 18) + 81 and (x - 9)(x - 9).

The expression describes the length of one side of the rooftop: x - 9

This is because the sides of a rectangle must be positive, and (x - 9) is a factor of (x + 9)² that represents a positive length.

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Find the value of this expression if x=8. x^2-8/x+1

Answers

The value of the expression when x=8 is 56/9.

To find the value of an expression, follow these steps:

Replace any variables in the expression with the given values. For example, if the expression is "3x + 5" and x = 2, replace x with 2 to get "3(2) + 5".Simplify the expression using the order of operations (PEMDAS/BODMAS). Evaluate any operations inside parentheses first, then perform any multiplications or divisions from left to right, and finally perform any additions or subtractions from left to right.Continue simplifying the expression until you reach a single value.

To find the value of the expression when x=8, we substitute 8 for x in the expression:

(8^2 - 8) / (8+1)

= (64 - 8) / 9

= 56/9

Therefore, the value of the expression when x=8 is 56/9.

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Find the smallest number of terms of the series ∑ n = 1 (-1)^n+1/2^n you need to be certain that the partial sum Sn is within 1/100 of the sum.n=2 n=4 n=6 n=8 n=7

Answers

We need at least 7 terms of the series to be certain that the partial sum Sn is within 1/100 of the sum.

We want to find the smallest value of n such that the absolute value of the difference between the sum of the first n terms and the sum of the entire series is less than 1/100.

The sum of the first n terms of the series is given by:

Sn = ∑_(k=1[tex])^n[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex]

We can write the sum of the entire series as:

S = ∑_(k=[tex]1)^∞[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex]

The absolute value of the difference between the sum of the first n terms and the sum of the entire series is:

|S - Sn| = |∑_(k=n+1[tex])^∞[/tex] [tex](-1)^(k+1)/2^k|[/tex]

We want to find the smallest value of n such that |S - Sn| < 1/100.

Let's start by evaluating the sum of the series:

S = ∑_(k=1) (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex] = 1/2 - 1/4 + 1/8 - 1/16 + ...

This is a geometric series with first term a = 1/2 and common ratio r = -1/2. The sum of the series is:

S = a/(1-r) = (1/2)/(1+1/2) = 1/3

Now we can write:

|S - Sn| = |∑_(k=n+1[tex])^∞[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k|[/tex] <= 1/[tex]2^(n+1)[/tex]

The last inequality is true because the terms of the series are decreasing in absolute value, and we are summing an infinite number of terms.

Therefore, we need to find the smallest value of n such that 1/2^(n+1) < 1/100. This gives:

n+1 > log2(100)

n > log2(100) - 1

n > 6.64

The smallest integer value of n that satisfies this inequality is n = 7.

Therefore, we need at least 7 terms of the series to be certain that the partial sum Sn is within 1/100 of the sum.

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Clarissa is cutting construction paper into rectangles for a project. She needs to cut one rectangle that is 10 inches × 14 1/2

inches. She needs to cut another rectangle that is 10 1/4

inches by 10 1/2 inches. How many total square inches of construction paper does Clarissa need for her project?

Answers

Clarissa needs a total of 251.25 square inches of construction paper for her project.

To find the total area of construction paper needed, we need to find the area of each rectangle and add them together.

The first rectangle has an area of 10 inches × 14.5 inches = 145 square inches.

The second rectangle has an area of 10.25 inches × 10.5 inches = 107.625 square inches.

Adding these two areas together, we get a total of 145 + 107.625 = 252.625 square inches.

Therefore, Clarissa needs a total of 251.25 square inches (rounded to two decimal places) of construction paper for her project.

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