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Q82.

Expert-verifiedFound in: Page 480

Book edition
7th Edition

Author(s)
James Stewart, Lothar Redlin, Saleem Watson

Pages
948 pages

ISBN
9781337067508

**Irrigation An irrigation system uses a straight sprinkler pipe **** long that pivots around a central point as shown. Because of an obstacle the pipe is allowed to pivot through ${\mathbf{280}}^{\mathbf{\xb0}}$ only. Find the area irrigated by this system.**

The area irrigated by this system is $\mathbf{219}\mathbf{,}\mathbf{911}{\mathbf{\text{ft}}}^{\mathbf{2}}$.

The area *A* of a sector with central angle of radians is:

$A=\frac{1}{2}{r}^{2}\theta $

So use above formula to find the area irrigated by this system.Given:Central angle $\left(\theta \right)={280}^{\xb0}$

radius $\left(r\right)=300ft$

To convert into radians, multiply the angle by $\frac{\pi}{180}$

To convert $\left(\theta \right)$ into radians, multiply by $\frac{\pi}{180}$:

$\theta ={280}^{\xb0}\times \frac{\pi}{{180}^{\xb0}}\text{rad}\phantom{\rule{0ex}{0ex}}=\frac{14\pi}{9}\text{rad}$

The area irrigated by this system is:

$A=\frac{1}{2}{r}^{2}\theta $

Put given values in the above formula, we get:

$A=\frac{1}{2}\times {300}^{2}\times \frac{14\pi}{9}$

Putting $\pi =3.14$ in the above formula, we get:

$A=\frac{1}{2}\times {300}^{2}\times \frac{14\times 3.14}{9}\phantom{\rule{0ex}{0ex}}=\frac{14\times 3.14\times 300\times 300}{18}\phantom{\rule{0ex}{0ex}}=\frac{3956400}{18}\phantom{\rule{0ex}{0ex}}\approx 219911.485751\phantom{\rule{0ex}{0ex}}\approx 219,911{\text{ft}}^{2}$

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