Work each problem. See Example 5. ...

Question

Work each problem. See Example 5.                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                      Irrigation Area A center-pivot irrigation system provides water to a sector-shaped field as shown in the figure. Find the area of the field if θ = 40.0° and r = 152 yd.

Accepted Answer

Welcome back. I am so glad you're here. We're told a person takes a part of a pizza in the shape given below, find the area of the portion of the pizza. If theta equals 35 degrees and R equals 10 centimeters, our answer choices are answer choice. A 1750 centimeters squared. Answer choice B 3500 centimeters squared. Answer choice C 61.1 centimeter squared and answer choice D 30.5 centimeters squared. Then we're given a diagram of a piece of pizza. Our piece of pizza is mostly triangular but one part of it is rounded. So actually this is a sector of a circle. So we've got our theta labeled for the inside angle of the sector of our circle. And then R is the length from that vertex to the outside edge of the circle. We recall from previous lessons that the formula for the area of a sector of a circle is a equals one half R squared theta. And this is great because we've got R and we've got theta one thing to note though, that's very important. Our angle, our theta is given in degrees. And we need to convert this to radiance before we solve. So we need to take our 35 degrees and multiply that by pi divided by 180 degrees that cancels out the degrees. And then we'll have a theta of seven pi divided by 36 makes sense. Pizza pie gotta have it in pi. All right. Now, let's fill this in for the area. We have one half multiplied by R squared. R is 10 centimeters. Both 10 and centimeters need to be squared multiplied by our theta in radiant is seven pi divided by 36. Now, we just need to simplify 10 centimeters squared is centimeters squared. We'll multiply that by one half. So we'll have 50 centimeters squared and then still multiplying that by seven pi divided by 36 plugging that in that simplifies to be 175 pi divided by 18 centimeters squared. But we want to convert this to just a number with a decimal and without pi in it. So this will round to be rounding it to the nearest 10th. We have 30. centimeters squared and we look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Work each problem. See Example 5. Irrigation Area A center-pivot irrigation system provides water to a sector-shaped field as shown in the figure. Find the area of the field if θ = 40.0° and r = 152 yd.

Key Concepts

Sector Area Formula

Radians vs. Degrees

Trigonometric Functions

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