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{(315)/(\pi )} \end{cases} \\\\\\ \cfrac{180\cdot 12.2}{\pi \sqrt{(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{(315)/(\pi )}\approx 10\\ \theta \approx 70 \end{cases}\implies A=\cfrac{70\cdot \pi \cdot 10^2}{360}`

Answer 2

The area of the garden and the length of the arc of the area used to grow zucchini indicates;

1. 10 feet

2. 63 feet

3. 61 square feet

The steps used to find the above values are as follows;

1. The length of the sprinkler, CA is equivalent to the radius of the circular garden

The area of the garden is 315 square feet, therefore;

π × The square of the radius of the garden = 315

π × (CA)² = 315

CA = √(315/π)

√(315/π) ≈ 10

The radius of the garden, CA ≈ 10 feet

2. The perimeter of the circular garden can be found using the formula for the circumference of a circle as follows;

The circumference of the circular garden = 2 × π × Radius of garden

Perimeter ≈ 2 × π × 10

2 × π × 10 ≈ 63

The perimeter of the garden is about 63 feet

3. The length of the arc DE ≈ 12.2 feet

The length of the arc, obtained from the circumference of the circular garden is; (θ/360) × 63 ≈ 12.2

θ ≈ 12.2 × 360/63

θ ≈ 69.7°

The area of the garden Mr. Clark used to grow Zucchini is therefore;

Area = (69.7°/360°) × 315

(69.7°/360°) × 315 ≈ 61

The area used to grow zucchini is about 61 square feet