Question 1137145
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.


<a href = "http://www.mesacc.edu/~scotz47781/mat120/notes/variation/joint/joint.html" target = "_blank">http://www.mesacc.edu/~scotz47781/mat120/notes/variation/joint/joint.html</a>


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.