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 = r + 5pi )

○ ( a = pi r + 2.5pi r )

○ ( a = pi r^2 + 6.25pi )

○ ( a = pi r^2 + 5pi r + 6.25pi )

Explanation:

Step1: Identify the radii

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

Step2: Calculate the area of the flower garden

The area of a circle is \( A = \pi r^2 \), so the area of the flower garden is \( \pi r^2 \).

Step3: Calculate the area of the entire round section

The area of the entire round section (with radius \( r + 2.5 \)) is \( \pi (r + 2.5)^2 \). Expanding this using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), we get:
\[

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

\]

Answer:

\( A = \pi r^2 + 5\pi r + 6.25\pi \) (the last option)