Question

1. the bottom of a circular swimming pool has an area of 200.96 square feet. what is the radius (in feet) of the swimming pool? use 3.14 for π. 22. the cost c (in dollars) of producing x widgets in represented by c = 4.5x². how many widgets are produced if the cost is $544.50? 23. two squares are drawn. the larger square has area of 400 square inches. the area of the two squares have a ratio of 1:4. what is the side length of the smaller square?

Explanation:

Question 21:

Step1: Recall the area formula for a circle

The area \( A \) of a circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle. We are given that \( A = 200.96 \) square feet and \( \pi = 3.14 \). We need to solve for \( r \).

Step2: Substitute the given values into the formula

Substitute \( A = 200.96 \) and \( \pi = 3.14 \) into the formula \( A = \pi r^2 \):
\[
200.96 = 3.14 \times r^2
\]

Step3: Solve for \( r^2 \)

Divide both sides of the equation by \( 3.14 \):
\[
r^2 = \frac{200.96}{3.14}
\]
Calculate the right-hand side:
\[
r^2 = 64
\]

Step4: Solve for \( r \)

Take the square root of both sides. Since the radius is a positive quantity, we consider the positive square root:
\[
r = \sqrt{64}
\]
\[
r = 8
\]

Step1: Recall the cost function

The cost \( C \) of producing \( x \) widgets is given by \( C = 4.5x^2 \). We are given that \( C = 544.50 \) dollars, and we need to solve for \( x \).

Step2: Substitute the given cost into the function

Substitute \( C = 544.50 \) into the formula \( C = 4.5x^2 \):
\[
544.50 = 4.5x^2
\]

Step3: Solve for \( x^2 \)

Divide both sides of the equation by \( 4.5 \):
\[
x^2 = \frac{544.50}{4.5}
\]
Calculate the right-hand side:
\[
x^2 = 121
\]

Step4: Solve for \( x \)

Take the square root of both sides. Since the number of widgets produced must be a positive integer, we consider the positive square root:
\[
x = \sqrt{121}
\]
\[
x = 11
\]

Step1: Recall the area ratio of squares

The ratio of the areas of two squares is equal to the square of the ratio of their corresponding side lengths. We are given that the ratio of the areas of the smaller square to the larger square is \( 1:4 \), and the area of the larger square is 400 square inches.

Step2: Find the area of the smaller square

Let the area of the smaller square be \( A_s \) and the area of the larger square be \( A_l = 400 \) square inches. The ratio \( \frac{A_s}{A_l} = \frac{1}{4} \). Solve for \( A_s \):
\[
A_s = \frac{1}{4} \times A_l
\]
Substitute \( A_l = 400 \):
\[
A_s = \frac{1}{4} \times 400
\]
\[
A_s = 100
\]

Step3: Find the side length of the smaller square

The area of a square is given by \( A = s^2 \), where \( s \) is the side length. For the smaller square, \( A_s = 100 \) square inches. Solve for \( s \):
\[
s^2 = 100
\]
Take the square root of both sides:
\[
s = \sqrt{100}
\]
\[
s = 10
\]

Answer:

The radius of the swimming pool is 8 feet.

Question 22: