pre calc why is my answer wrong?

Question

Ramogi has a sprinkler that has several varieties of coverage. The 42 degree sprinkler head sprays water a distance of up to 15.6 feet from the head. What area will be covered by the spray of the 42 degree sprinkler head to the nearest square foot?

I used s=r*theta is radians

15,6*42*pie/180

i get 11.44 but this is wrong am I using the wrong formula?

Answer 1

Hi Samantha, you are using the wrong formula. s = r*theta is for the arc length but you're asked to find the area covered by the portion of a circle.

Depending on whether θ is in degrees or radians, there are two formulas for the area of the sector (portion) of a circle:

if θ is in radians: Area = (1/2)* r²*θ

if θ is in degrees: Area = (π/360)* r²*θ

(I'd recommend only memorizing one formula PLUS knowing how to convert from radians to degrees and vice-versa, but if you prefer memorizing both, you sure can.)

Now, let's use the 2nd formula: Area = (π/360)* 15.6²*42 = 89.19609862 square foot.

Round that to the nearest square foot and you're all set.