A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of $$ 150^{\circ} $$. What area of lawn receives water?

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UpStudy AI · Accepted Answer

To find the area of the lawn that receives water from the sprinkler, we can model the area as a sector of a circle. The formula for the area $$ A $$ of a sector is given by:

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

where:
- $$ r $$ is the radius (the distance the sprinkler sprays water),
- $$ \theta $$ is the angle in radians.

### Step 1: Convert the angle from degrees to radians

The angle in radians can be calculated using the conversion:

$$
\theta = \frac{\pi}{180} \times \text{degrees}
$$

For $$ 150^{\circ} $$:

$$
\theta = \frac{\pi}{180} \times 150
$$

### Step 2: Calculate the area

Now we can substitute $$ r = 30 $$ feet and the calculated $$ \theta $$ into the area formula.

Let's perform these calculations step by step.

1. Convert $$ 150^{\circ} $$ to radians.
2. Calculate the area using the sector area formula.

I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\pi }{180}\times 150$$
- step1: Reduce the numbers:
  $$\frac{\pi }{6}\times 5$$
- step2: Multiply:
  $$\frac{\pi \times 5}{6}$$
- step3: Multiply:
  $$\frac{5\pi }{6}$$
Calculate or simplify the expression $$ (1/2)*(30^2)*(\pi/180)*150 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{2}\times 30^{2}\times \frac{\pi }{180}\times 150$$
- step1: Multiply the terms:
  $$\frac{5\pi }{12}\times 30^{2}$$
- step2: Rewrite the expression:
  $$\frac{5\pi }{12}\times 6^{2}\times 5^{2}$$
- step3: Rewrite the expression:
  $$\frac{5\pi }{6\times 2}\times 6^{2}\times 5^{2}$$
- step4: Reduce the numbers:
  $$5\pi \times 3\times 5^{2}$$
- step5: Rewrite the expression:
  $$5\pi \times 75$$
- step6: Multiply the terms:
  $$375\pi $$
### Step 1: Convert the angle to radians

The angle $$ 150^{\circ} $$ in radians is:

$$
\theta \approx 2.617994 \text{ radians}
$$

### Step 2: Calculate the area of the sector

Using the formula for the area of the sector:

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

Substituting $$ r = 30 $$ feet and $$ \theta \approx 2.617994 $$:

$$
A \approx \frac{1}{2} \times 30^2 \times 2.617994
$$

Calculating this gives:

$$
A \approx 375\pi \text{ square feet} \quad \text{or} \quad A \approx 1178.10 \text{ square feet}
$$

### Final Answer

The area of the lawn that receives water from the sprinkler is approximately $$ 1178.10 $$ square feet.
The area of the lawn that receives water is approximately 1178.10 square feet.

A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of . What area of lawn receives water?

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