Here's Question 39 for my review

Question

The horsepower that can be safely transmitted to a shaft varies jointly as the shaft's angular speed of rotation (in revolutions per minute) and the cube of its radius. A 2-inch radius shaft making 120 revolutions per minute safely transmits 40 horsepower. Find how much horsepower can be safely transmitted by a 3-inch radius shaft making 80 revolutions per minute. (Variation)

Answer 1

Let H=horsepower, A = angular speed in rotations per minute, r = radius, and K be the constant varialble that jointly varies the units together.

The formula for this joint variation problem would be

H= KAR^3

Plugging in the units and solving for the constant would be

40=K(120)(2)^3

40=K(120)(8)

40=960

1/24=K

Now using the fact that K = 1/24 plug in the know units and solve it again

H=(1/24)(80)(3)^3

H= (1/24)(80)(27)

H=90

Therefore it would take 90 Horsepower to safely transmit an 3 inch shaft rotating at 80 revolutions per min.