Question 1137145: Would someone please help explain how to correctly solve this? Thank you.
The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm)) and the cube of the diameter. If the shaft of a certain material 4 inches in diameter can transmit 40 hp at 120 rpm, what must the diameter, d, be in order to transmit 65 hp at 140 rpm? (Round your answer to two decimal places.)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the joint variation equation is z = k * x * y.
in your problem:
z = the horsepower that a shaft can safely transmit.
x is the speed in revolutions per minute.
y is the cube of the diameter.
if the shaft is 4 inches in diameter, then y = 4^3 = 64.
if the revolutions per minute are 120, then x = 120.
if the horsepower is 40, then z = 40.
the equation becomes 40 = k * 120 * 64.
solve for k to get k = 40 / (120 * 64) =.0052083333.
confirm by replacing k with that to get 40 = .005208333 * 120 * 64 = 40, confirming that the value of k is good.
k is the constant of variation and remains the same in all problems involving it.
when z = 65 and x = 140, the equation becomes 65 = k * 140 * y which becomes 65 = .005208333 * 140 * y
solve for y to get y = 65 / (.005208333 * 140) = 89.14285714.
y is equal to the cube of the diameter, so the diameter would then be cube root of (89.14285714) = 4.46713266 inches.
in order to get 65 horsepower at 140 rpm safely, the diameter of the shaft must be 4.46713266 inches.
the method is to first solve for k with the given information and then to use that value of k to solve for the variable in the equation that is unknown.
you solve for k with z = 40 and x = 120 and y = d^3 = 4^3 = 64.
you then used k to solve for y when z = 65 and x = 140.
since y = d^3, you then needed to get the cube root of y to find d.
d is the diameter of the shaft.
here's a reference on joint variation.
http://www.mesacc.edu/~scotz47781/mat120/notes/variation/joint/joint.html
using that reference, you would replace z with h for the horsepower and x with r for rpm and y with d^3 for the cube of the diameter.
the formula would then become h = k * r * d^3.
when h = 40 and r = 140 and d = 4, the equation would become 40 = k * 120 * 64.
you'll get the same answer doing it this way instead of the way i showed you.
both ways are equivalent.
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