Question

1. find the total area of a round section of land that has a 2.5-foot-wide enclosure of pebbles surrounding a round garden of flowers.

○ ( a = pi r + 2.5pi r )

○ ( a = pi r^2 + 6.25pi )

○ ( a = pi r^2 + 5pi r + 6.25pi )

○ ( a = r + 5pi )

Explanation:

Step1: Identify the radius of the total circle

Let the radius of the flower garden be \( r \). The pebble enclosure is 2.5 feet wide, so the radius of the total round section (garden + pebbles) is \( R = r + 2.5 \).

Step2: Recall the area formula for a circle

The area of a circle is \( A=\pi R^{2} \), where \( R \) is the radius.

Step3: Substitute \( R = r + 2.5 \) into the area formula

\[

$$\begin{align*} A&=\pi(r + 2.5)^{2}\\ &=\pi(r^{2}+2\times r\times2.5+(2.5)^{2})\\ &=\pi(r^{2}+5r + 6.25)\\ &=\pi r^{2}+5\pi r+6.25\pi \end{align*}$$

\]

Answer:

\( A = \pi r^{2}+5\pi r + 6.25\pi \) (the third option)