Question

Devon’s bike has wheels that are 27 inches in diameter. After the front wheel picks up a tack, Devon rolls another 100 feet and stops. How far above the ground is the tack?

Answer 1

100 ft = 1200 inches
one rotation covers 27π or 84.823 inches
so the wheel went 1200/84.823 or 14.1471 rotations

We are interested in the .1471 part of the last rotation.
make a sketch of the circle, draw in vertical radius, and label an arc length of .1471(84.823) or 12.478 inches.
We need the central angle which subtends that arc of 12.478
from arc = rØ
we have
12.478 = 27Ø
Ø = .46215 radians

From the end of the arc, draw a horizontal cutting the radius to create a right-angled triangle
let h be the length form the centre of the circle to that horizontal

cos .46215 = h/27
h = 27cos.4615 = 24.16

so the nail is 27 - 24.16 or 2.8 inches above the ground.

You better check my arithmetic.