Question

pre-ap algebra 2 lesson 2.1 exit ticket
a circular garden with a radius of 15 feet is to be covered with topsoil at a cost of $1.25 per square foot of garden space.
(a) what formula do you need to determine the area of the circular garden?
(b) determine the area of this garden to the nearest square foot. show your work.
(c) using your answer from part (b), calculate the cost of covering the garden with topsoil. show your work.
(d) write a formula for the cost of covering the garden with topsoil as a function of the radius of the garden.

Explanation:

Part (a)

Step1: Recall circle area formula

The formula for the area \( A \) of a circle with radius \( r \) is \( A = \pi r^2 \). This is the standard formula used to find the area of a circular region.

Step1: Identify given values

The radius \( r = 15 \) feet. The formula for the area of a circle is \( A=\pi r^2 \).

Step2: Substitute \( r = 15 \) into the formula

Substitute \( r = 15 \) into \( A=\pi r^2 \), we get \( A=\pi\times(15)^2 \). Calculate \( 15^2 = 225 \), so \( A = 225\pi \).

Step3: Approximate the value

Using \( \pi\approx3.1416 \), then \( A\approx225\times3.1416 = 706.86 \). Rounding to the nearest square foot, we get \( 707 \) square feet.

Step1: Identify cost per square foot

The cost per square foot is \( \$1.25 \), and from part (b) the area is approximately \( 707 \) square feet.

Step2: Calculate total cost

Total cost \( C=\text{Cost per square foot}\times\text{Area} \). So \( C = 1.25\times707 \).

Step3: Compute the product

\( 1.25\times707 = 883.75 \).

Answer:

The formula is \( A = \pi r^2 \) (where \( A \) is the area and \( r \) is the radius of the circle).

Part (b)