1. a forest fire has been burning for several days. the burned area, in acres, is given by
the equation y =(4,800) 24, where d is the number of days since the area of the
fire was first measured.
a. complete the table.
d, days since first
measurement
y, acres burned
since fire started
b. look at the value of y = 4,800 - 20
when d = -1. what does it tell you
about the area burned in the fire?
what about when d = -3?
4800
0
-1
2400
-2
1200
-3
600
-5
150
c. how much area had the fire burned
a week before it measured 4,800
acres? explain your reasoning.

Answer:

The perfect factored square trinomial that is equivalent to the expression x^2 + 8x + 16 is:

(x + 4)^2

To see why this is the case, you can expand the expression (x + 4)^2 using the FOIL method:

(x + 4)^2 = (x + 4) * (x + 4)

= x^2 + 4x + 4x + 16

= x^2 + 8x + 16

So, x^2 + 8x + 16 can be factored as (x + 4)^2, which is a perfect square trinomial.

Answer:

The equation of a circle with center (a,b) and radius r is given by:

(x - a)² + (y - b)² = r²

Comparing this to the equation x² + y² = 12², we can see that the center of the circle is (0,0) and the radius is 12. Therefore, the radius of the circle is 12 units.

The number of favorable outcomes for spinning the same color twice will depend on the number of colors on the spinner.

If there are only two colors on the spinner, such as red and blue, then there will be only one favorable outcome, which is spinning either red or blue twice.

If there are more than two colors on the spinner, the number of favorable outcomes will depend on the number of times each color appears on the spinner.

For example, if there are four colors on the spinner, and each color appears equally, then there will be four favorable outcomes: spinning red twice, spinning blue twice, spinning green twice, or spinning yellow twice.

In general, if there are n colors on the spinner and each color appears with equal probability, then the number of favorable outcomes for spinning the same color twice will be n.

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Answer:

since 2x = z

replace z with 2x

900000 = x+y+z

900000 = x+y+2x

900000 = 3x+y - eqn (1)

79750= 0.08x +0.09y+0.01z

79750 = 0.08x +0.09y+0.01(2x)

79750 = 0.08x+0.09y+0.02x

79750 = 0.10x +0.09y - eqn(2)

from eqn(1)

900000 = 3x + y

y = 900000-3x - eqn(3)

substitute eqn(3) in eqn(2)

79750 = 0.1x +0.09y

79750=0.1x + 0.09(900000-3x)

79750=0.1x+ 81000 - 0.27x

collect like terms

79750 -81000 = 0.1x-0.27x

-1250 = -0.17x

to find x divide both sides by -0.17

x = -1250/-0.17 ~= 7353

since 2x = z

2*7353 = 14706

in eqn(3)

y = 900000-3x

y= 900000-3(7353)

y = 900000-22059

y = 877941

x =7353,y= 877941,z=14706

Using  trignometric functions the shorter side AD has length a ≈ 25.75 cm and the longer side AB has length b ≈ 34.25 cm.

In the given scenario, we have a rectangle with sides AD and AB. The length of AD is represented as 'a' and is approximately 25.75 cm, while the length of AB is denoted as 'b' and is approximately 34.25 cm. The diagonal AC of the rectangle has a length of 42.3 cm and forms an angle of 53° with AD.

To find the lengths of sides a and b, we can utilize trigonometric functions, specifically cosine and sine. Since we have the length of the diagonal AC and the angle it forms with AD, we can set up the following equations:

cos(53°) = a/42.3

sin(53°) = b/42.3

By rearranging the equations, we can solve for a and b:

a = 42.3 * cos(53°) ≈ 25.75 cm

b = 42.3 * sin(53°) ≈ 34.25 cm

By substituting the given values into the equations, we can determine that the length of AD (a) is approximately 25.75 cm, and the length of AB (b) is approximately 34.25 cm.

These calculations allow us to find the side lengths of the rectangle based on the given information about the diagonal length and angle. Understanding trigonometric relationships enables us to solve geometric problems involving angles, sides, and diagonals in various shapes and configurations.

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The Mean Value Theorem can be used to prove that the square root of 6a+31 lies between two values, where one value is equal to the function evaluated at a divided by the square root of 6, and the other value is equal to the function evaluated at a plus one divided by the square root of 6.

Let f(x) = √(6x + 31) and choose any value of a such that a > -31/6.

By the Mean Value Theorem, there exists some c in (a, a+1) such that:

f(a+1) - f(a) = f'(c)

where f'(c) is the derivative of f(x) evaluated at c.

We have:

f'(x) = 3/√(6x+31)

Thus, we can write:

f(a+1) - f(a) = (3/√(6c+31)) * (a+1 - a)

Simplifying, we get:

f(a+1) - f(a) = 3/√(6c+31)

Since a < c < a+1, we have:

a < c

√(6a+31) < √(6c+31)

√(6a+31) < (3/√(6c+31)) * √(6c+31)

√(6a+31) < f(a+1) - f(a)

Therefore, we can write:

f(a) < √(6a+31) < f(a+1)

f(a) = √(6a + 31)/√6

f(a+1) = √(6(a+1) + 31)/√6

Substituting these values, we get:

(√(6a + 31))/√6 < √(6a+31) < (√(6(a+1) + 31))/√6

Simplifying, we get:

√(6a + 31)/√6 < √(6a+31) < √(6a + 37)/√6

Hence, we have shown that the square root of 6a+31 lies between two values, where one value is equal to the function evaluated at a divided by the square root of 6, and the other value is equal to the function evaluated at a plus one divided by the square root of 6.

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The angle between u and v is approximately 104.66 degrees. To find the angle between two vectors u and v, we can use the dot product formula:

cos(theta) = (u · v) / (||u|| ||v||)

where ||u|| and ||v|| are the magnitudes of u and v, respectively.

First, let's compute the dot product of u and v:

u · v = [tex](4)(5) + (3)(-12) = 20 - 36 = -16[/tex]

Next, we need to find the magnitudes of u and v:

[tex]||u||[/tex] = sqrt([tex]4^2[/tex] + [tex]3^2[/tex]) = 5

[tex]||v||[/tex] = sqrt([tex]5^2[/tex] + (-12[tex])^2[/tex]) = 13

Now we can substitute these values into the formula for cos(theta):

cos(theta) = [tex](-16) / (5 * 13) = -0.246[/tex]

To find the angle theta, we take the inverse cosine of cos(theta):

theta = [tex]cos^-1[/tex](-0.246) = 104.66 degrees

Therefore, the angle between u and v is approximately 104.66 degrees.

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(A) The length that beats 80 seconds is 256 cm.

(B) The time of a pendulum with length 36 cm is 30 seconds.

(A) According to the given information, the time of a pendulum varies as the square root of its length. Let's denote time as T and length as L. Therefore, T ∝ √L. To find the constant of proportionality, we can use the provided data: T1 = 15 seconds and L1 = 9 cm. So, we have T1 / √L1 = k, where k is the constant. Now, let's find k: k = 15 / √9 = 15 / 3 = 5.

Now, we want to find the length (L2) of a pendulum that beats 80 seconds (T2). We can use the formula T2 = k * √L2. Substituting the values, we get 80 = 5 * √L2. To find L2, we can rearrange and solve for it: L2 = (80 / 5)² = 16² = 256 cm.

(B) To find the time (T3) of a pendulum with a length of 36 cm (L3), we can use the same formula with the known constant k: T3 = k * √L3. Substituting the values, we get T3 = 5 * √36 = 5 * 6 = 30 seconds.

In conclusion, the length of a pendulum that beats 80 seconds is 256 cm, and the time of a pendulum with a length of 36 cm is 30 seconds.

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Ted spent 67 minutes reading.

First, let's determine how long Jared spent reading:

Jared = Pete - 52 minutes

Jared = 3 hours * 60 minutes/hour - 52 minutes

Jared = 148 minutes

Now we can use the fact that Ted spent 1 hour 21 minutes less than Jared:

Ted = Jared - 1 hour 21 minutes

Ted = 148 minutes - 81 minutes

Ted = 67 minutes

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The regression line equation, can be found to be y = 0.90x - 3.79

Find the slope using the slope formula :

m = ( 5 x 1944 - 98 x 69 ) / ( 5 x 2580 - 98² )

m = ( 9720 - 6762 ) / ( 12900 - 9604 )

m = 2958 / 3296

=  0.8975

Then find the y - intercept :

b = ( 69 - 0. 8975 x 98) / 5

b = ( 69 - 87. 945) / 5

b = - 18. 945 / 5

= - 3.789

The regression equation is:

y = 0.90x - 3.79

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The perimeter of the rectangle is represented by the expression 4s - 10.

How to calculate perimeter of a rectangle?

To calculate the perimeter of a rectangle, you need to add up the lengths of all four sides.

In the problem given, we know that the rectangle is 6 inches shorter than the square and 1 inch longer.

Let's call the length of the rectangle l and the width w.

We know that the length of the square is equal to its width (since it's a square), so the length of the rectangle must be l = s - 6, and the width must be w = s + 1.

To find the perimeter, we add up all four sides: P = 2l + 2w = 2(s-6) + 2(s+1) = 4s - 10.

Therefore, the expression that represents the perimeter of the rectangle is 4s - 10.

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Answer is 9.5 tons of cement

Anne's Road Paving Company initially mixed 16 1/4 tons of cement. They used 6 3/4 tons for paving a street downtown. To find the remaining amount of cement, subtract the used amount from the initial amount:

16 1/4 - 6 3/4 = 15 1/4 - 5 3/4 = 9 1/2 tons.

So, they had 9 1/2 tons of cement left.

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The maximum height the golf ball reached before landing back on the ground is 3.75 yards.

To find the maximum height the golf ball reached before landing back on the ground, we need to find the vertex of the quadratic function[tex]h(t) = -0.6t^2 + 3t.[/tex] The vertex of a quadratic function in the form of[tex]f(x) = ax^2 + bx + c[/tex] is given by the formula x = -b/(2a).

In this case, a = -0.6 and b = 3. Plugging these values into the formula:

t = -3 / (2 * -0.6) = 3 / 1.2 = 2.5

Now that we have the time at which the ball reaches its maximum height, we can plug this value back into the height function to find the maximum height:

[tex]h(2.5) = -0.6(2.5)^2 + 3(2.5) = -0.6(6.25) + 7.5 = -3.75 + 7.5 = 3.75[/tex]

So, the maximum height the golf ball reached before landing back on the ground is 3.75 yards.

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For this above box plot prompt, the answer are given below.

Part A

From the box plots, we can see that the Red Team has the least variability and spread of times, followed by the Blue Team and then the Green Team.

The Red Team's box is the smallest, indicating that their times are more tightly clustered together.

Blue Team:

Q1: 75

Q2: 82

Q3: 87

IQR: 12

Upper fence: Q3 + 1.5IQR = 87 + 1.512 = 105

There are no outliers

Green Team:

Q1: 70

Q2: 75

Q3: 80

IQR: 10

Upper fence: Q3 + 1.5IQR = 80 + 1.510 = 95

There is one outlier at 90

Red Team:

Q1: 80

Q2: 83

Q3: 87

IQR: 7

Upper fence: Q3 + 1.5IQR = 87 + 1.57 = 98.5

There are no outliers

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The absolute maximum occurs at x = -2 and the absolute minimum occurs at x = 0 and x = 2 and  The absolute maximum is at x = -2 and the absolute minimum is at x = 0, 2.
a. The local minimum is at x=2 and there is no local maximum.
b. The local maximum is at x=1 and the local minimum is at x=-2 and x=2.
c. The absolute maximum is at x=0 and the absolute minimum is at x=2.
(Note: To find the absolute maximum and minimum, we need to evaluate the function at the critical points and endpoints of the interval. The critical points are x=0 and x=2, and the endpoints are x=-2 and x=2. The absolute maximum is the largest value among these, which is f(0)=0. The absolute minimum is the smallest value among these, which is f(2)=-4.)
Given the function f(x) = x²(x - 2) on the interval [-2, 2]:

A. To find the local minima and maxima, we need to take the first derivative and find its critical points.

f'(x) = 3x² - 4x
Solving for x, we get x = 0 and x = 4/3.

However, x = 4/3 is not within the interval [-2, 2], so the only critical point within the interval is x = 0.

There is a local minimum at x = 0, and no local maximum. Therefore, the answer is:

A. The local minimum is at x = 0 and there is no local maximum. (Type an integer or a simplified fraction.)

B. For the absolute maximum and minimum, we need to evaluate the function at the endpoints and the critical point within the interval.

f(-2) = (-2)²(-2 - 2) = 16
f(0) = (0)²(0 - 2) = 0
f(2) = (2)²(2 - 2) = 0

The absolute maximum occurs at x = -2 and the absolute minimum occurs at x = 0 and x = 2. The answer is:

B. The absolute maximum is at x = -2 and the absolute minimum is at x = 0, 2. (Use a comma to separate answers as needed. Type integers or simplified fractions.)

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The amount each of them pays when their electricity bill is $110, $165, and $352 respectively, in the ratio 2:4:5, are:

1) $20, $40, $50

2) $30, $60, $75

3) $64, $128, $160

To find out how much each of them pays, we'll use the given ratio of 2:4:5 and divide the total bill among them accordingly.

Total bill: $110

The total ratio is 2+4+5=11.

Greg's share: (2/11) * $110 = $20

Harry's share: (4/11) * $110 = $40

Ian's share: (5/11) * $110 = $50

Therefore, Greg pays $20, Harry pays $40, and Ian pays $50.

Total bill: $165

The total ratio is still 2+4+5=11.

Greg's share: (2/11) * $165 = $30

Harry's share: (4/11) * $165 = $60

Ian's share: (5/11) * $165 = $75

Therefore, Greg pays $30, Harry pays $60, and Ian pays $75.

Total bill: $352

The total ratio remains the same: 2+4+5=11.

Greg's share: (2/11) * $352 = $64

Harry's share: (4/11) * $352 = $128

Ian's share: (5/11) * $352 = $160

Therefore, Greg pays $64, Harry pays $128, and Ian pays $160.

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The loan future value is $7726.73.

To find the loan future value, we need to calculate the total amount that Robert will owe at the end of the 2-year loan term, including both the principal (initial loan amount) and the interest.

To begin, we can use the formula for calculating compound interest:

[tex]A = P(1 + r/n)^{(nt)[/tex]

where A is the final amount, P is the principal, r is the interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

In this case, we know that the principal is $7200, the interest rate is 4.3% (or 0.043 as a decimal), the loan term is 2 years, and the interest is compounded once per year (n = 1).

Substituting these values into the formula, we get:

A = 7200(1 + 0.043/1)²

A = 7200(1.043)²

A = 7726.73

Therefore, the loan future value is $7726.73. This means that at the end of the 2-year loan term, Robert will owe a total of $7726.73, which includes the original $7200 loan amount and $526.73 in interest.

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Yes, arcsin and sin^-1 both represent the inverse sine function.

process of finding inital speed:

a) To find the ball's initial speed, we can use the range formula for projectile motion:
R = (v₀² * sin(2θ)) / g

where R is the range (10 m),

v₀ is the initial speed,

θ is the launch angle (45 degrees), and

g is the acceleration due to gravity (9.81 m/s²).

We can solve for v₀:

10 = (v₀² * sin(90)) / 9.81
10 = (v₀²) / 9.81
v₀² = 10 * 9.81
v₀ = sqrt(10 * 9.81)

The ball's initial speed is sqrt(10 * 9.81) m/s.

b) For the same initial speed, we can find the two firing angles that make the range 6 m:

6 = (v₀² * sin(2θ)) / 9.81
Now, we can use the initial speed found in part (a):

6 = (10 * 9.81 * sin(2θ)) / 9.81
0.6 = sin(2θ)

To find the two angles, we can use the arcsin function:

θ₁ = 1/2 * arcsin(0.6)
θ₂ = π - 1/2 * arcsin(0.6)

The two firing angles are 1/2 * arcsin(0.6) and π - 1/2 * arcsin(0.6).Yes, arcsin is the same as sin^(-1);

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The expression that represents a strategy for estimating the farm's output for 2015 is 1.153w, where w is the farm's wheat output in 2013.

Let w be the farm's wheat output in 2013. We know that the output increased by 9.8% from 2013 to 2014, so the output in 2014 can be estimated as:

w + 0.098w = 1.098w

This expression represents a strategy for estimating the farm's output for 2014.

Similarly, the output in 2015 can be estimated as:

(1.098w) + 0.051(1.098w) = (1 + 0.051)(1.098w)

Simplifying this expression, we get:

1.153w

Therefore, the expression that represents a strategy for estimating the farm's wheat output for 2015 is:

1.153w

where w is the farm's wheat output in 2013 (i.e., 19.8 tons).

So we can estimate the farm's wheat output in 2015 as:

1.153(19.8) = 22.82 tons (rounded to two decimal places)

Note that this is only an estimate, based on the assumption that the percentage increases from 2013 to 2014 and from 2014 to 2015 will continue to hold in the future.

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Val's calculation of 1,787.52 m² is incorrect.

What is area of semicircle?

The area of a semicircle is half the area of the corresponding circle. If r is the radius of the semicircle, then the area of the semicircle is:

A(semicircle) = (1/2) π r²

To find the area enclosed by the figure, we need to add the areas of the square and the four semicircles.

The area of the square is:

[tex]A_{square}[/tex] = (56 m)² = 3,136 m²

The area of one semicircle is half the area of the corresponding circle, and the radius of the circle is equal to the side length of the square. Therefore, the area of one semicircle is:

[tex]A_{semicircle}[/tex] = (1/2) π (56/2)²= 1,554.56 m²

The total area enclosed by the figure is:

[tex]A_{total}[/tex] = [tex]A_{square}[/tex]+ 4 [tex]A_{semicircle}[/tex] = 3,136 + 4(1,554.56) = 9,901.44 m²

Therefore, Val's calculation of 1,787.52 m² is incorrect.

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Question:

Val needs to find the area enclosed by the figure. The figure is made by attaching semicircles to each side of a 56m​-by-56m square. Val says the area is 1,787.52m. Find the area enclosed by the figure. Use 3.14 for π. What error might have Val made?

The postulate or theorem that can be used to prove that ΔABC ≅ ΔDCB is the "Side-Side-Side (SSS) theorem".

Hence, the correct option is A.

Since in both triangles ΔABC and ΔDCB, we have

Therefore, by SSS theorem, we can conclude that ΔABC ≅ ΔDCB.

Hence, the correct option is A.

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The probability that a randomly selected balloon animal is green and shaped like a dog is 0.231.

The probability is found using the data table given below:

Purple and giraffe = 3; Purple and dog =  3; Green and giraffe = 4; Green and dog = 3

Out of the total number of balloon animals made, the number of green dog balloon animals is 3.

The probability of randomly selecting a green dog balloon animal is found using the formula:

Probability = 3 / (3 + 3 + 4 + 3) = 3/13

Probability = 0.231

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The width of the gravel pathway is 7 meters.

The length of the rectangular garden is 50m and the width is 13m. Let's assume the width of the gravel pathway to be w meters.

The length of the rectangular garden including the two widths of the pathway would be 50+2w meters, and the width including the two widths of the pathway would be 13+2w meters.

The area of the rectangular garden including the pathway is the product of the length and the width:

(50+2w)(13+2w)

We can now set up an equation using the area of the garden and the amount of gravel available:

(50+2w)(13+2w) - 50*13 = 80

Simplifying this equation gives:

4w^2 + 126w - 3196 = 0

This is a quadratic equation that we can solve for w using the quadratic formula:

w = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 4, b = 126, and c = -3196.

Plugging in these values and solving for w gives:

w = 7 or w = -22.75

Since the width of the pathway cannot be negative, the only valid solution is w = 7.

Therefore, the width of the gravel pathway is 7 meters.

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The cube root of the given function is [tex]x=2\sqrt[3]{2}[/tex].

The given function is x³=16.

Here, the given function can be written as

[tex]x=\sqrt[3]{16}[/tex]

[tex]x=\sqrt[3]{2\times2\times2\times2}[/tex]

[tex]x=\sqrt[3]{2^3\times2}[/tex]

[tex]x=2\sqrt[3]{2}[/tex]

Therefore, the cube root of the given function is [tex]x=2\sqrt[3]{2}[/tex].

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"Your question is incomplete, probably the complete question/missing part is:"

How do you solve the cube root function of x³=16.

The spindle speed for a cut taken with a 4-inch high-speed drill and a cutting speed of 30 feet per minute is approximately 35.1 revolutions per minute.

Speed is a measure of how fast an object is moving. It is usually measured in units of distance per unit time, such as miles per hour or meters per second. Speed is an important concept in physics, engineering, and everyday life

We can use the formula for spindle speed that relates spindle speed to cutting speed and tool diameter:

S = (C × 12) / (π × D)

where S is spindle speed, C is cutting speed in feet per minute, D is tool diameter in inches, and π is the mathematical constant pi.

We know that for a 3-inch high-speed drill with a cutting speed of 70 feet per minute, the spindle speed is 88.2 revolutions per minute. We can use this information to solve for the constant of proportionality k:

88.2 = (70 × 12) / (π × 3)

k = 88.2 × (π × 3) / (70 × 12)

k ≈ 0.0039

Now we can use the value of k to find the spindle speed for a 4-inch high-speed drill with a cutting speed of 30 feet per minute:

S = k × C × 12 / D

S = 0.0039 × 30 × 12 / 4

S = 35.1

Therefore, the spindle speed for a cut taken with a 4-inch high-speed drill and a cutting speed of 30 feet per minute is approximately 35.1 revolutions per minute.

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To determine the best estimation for centering the dresser along the wall, we need to consider the length of the wall and the length of the dresser. Let's call the length of the wall "W" and the length of the dresser "D".

Since we don't know the actual values of W and D, we'll have to work with the given options.

Option A suggests placing the dresser about 6 feet from each end of the wall. This would leave a space of W - 12 feet in the middle of the wall, which would be the total space available for the dresser to be centered.

Option B suggests placing the dresser about 8 feet from each end of the wall. This would leave a space of W - 16 feet in the middle of the wall, which would be the total space available for the dresser to be centered.

Option C suggests placing the dresser about 10 feet from each end of the wall. This would leave a space of W - 20 feet in the middle of the wall, which would be the total space available for the dresser to be centered.

Option D suggests placing the dresser about 12 feet from each end of the wall. This would leave a space of W - 24 feet in the middle of the wall, which would be the total space available for the dresser to be centered.

To find the best estimation for centering the dresser along the wall, we need to determine which option provides the closest match between the available space in the middle of the wall and the length of the dresser.

Without knowing the actual values of W and D, it's difficult to say for certain which option is best. However, we can make an educated guess by considering the lengths of typical bedroom walls and dressers.

Based on this, option C (placing the dresser about 10 feet from each end of the wall) seems like a reasonable estimation for centering the dresser along the wall. This option provides a space of W - 20 feet in the middle of the wall, which is likely sufficient for most dressers.

Of course, the actual placement of the dresser will depend on other factors as well, such as the layout of the room and the location of other furniture. It's always a good idea to measure carefully and test different arrangements before settling on a final placement for any piece of furniture.

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Answer: around 2.6 cm because I rounded to the tenth.

Step-by-step explanation:

r^2=170/8.1×3.14

r^2=170/25.434

r^2≈6.68

Next square root both sides so r^2 becomes r and 6.68 square rooted is about 2.6 cm is the radius.

R≈2.6cm

Andres brought the most milk, and Carlos brought the least milk.

To find out the student who bought the most milk, you need to convert the liters decimals so that they can be compared evenly.

Andrés brought 2²

= 2 x 2

= 4 liters of milk.

Bruno brought 13/4:

= 13 / 4

= 3.25 liters of milk.

Carlos bought 1. 16 liters and Daniel bough 1. 3 liters.

This shows that Andres bought the most milk and Carlos bought the least amount.

Find out more on liters at https://brainly.com/question/17433466

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To find the cat's final position, we need to add up all the vectors representing the cat's movements.

The cat first travels along the vector −1, 2.

Next, the cat chases the ball along the vector 2, 6 − , which we can write as (2, 6) − (0, 1) = (2, 5).

Then, the cat darts after the ball 1.5 times along the vector 4, 3, which we can write as 1.5(4, 3) = (6, 4.5).

Finally, the cat's position after catching the ball is the sum of all these vectors:

(-1, 2) + (2, 5) + (6, 4.5) = (7, 11.5)

Therefore, the vector describing the cat's final position is (7, 11.5).