Question

(All problems are connected with eachother.) Directions: Mr. Clark's backyard vegetable garden is circular, centered at point A, and uses a small pivot irrigation system. A long sprinkler, marked by the line CA, rotates around the fixed point A in order to water the garden evenly. Every point in the figure represents a single post that Mr. Clark uses for measurement.
The total area of the garden is approximately 315 square feet.

1. To the nearest foot, how long is the sprinkler?


2. To the nearest foot, approximately how many feet of fencing would be needed to place a small fence on the perimeter of the garden?

3. If Mr. Clark measured arc DE to be about 12.2 feet long, then how many square feet of his garden is used to grow zucchini?

Answer 1

Check the picture below. part 1 and 2 are there.

part 3)

$\bf \textit{arc's length}\\\\ s=\cfrac{r\theta \pi }{180}\implies \cfrac{180s}{r\pi }=\theta \qquad \begin{cases} \theta =\textit{angle in degrees}\\ r=radius\\ ----------\\ s=12.2\\ r=\sqrt{\frac{315}{\pi }} \end{cases} \\\\\\ \cfrac{180\cdot 12.2}{\pi \sqrt{\frac{315}{\pi }}}=\theta \implies 69.8075^o\approx \theta \implies 70^o\approx\theta$

$\bf \textit{area of a sector of a circle}\\\\ A=\cfrac{\theta \pi r^2}{360}\quad \begin{cases} r=\sqrt{\frac{315}{\pi }}\approx 10\\ \theta \approx 70 \end{cases}\implies A=\cfrac{70\cdot \pi \cdot 10^2}{360}$