The efficiency E of a screw is given by E=T(1-0.35T)/T+0.35
where T is the tangent of the pitch angle of the screw and 0.35 is the coefficient of friction . For what value of T>0 is the efficiency the greatest?
just a normal min/max problem.
E = T(1-0.35T)/T + 0.35
you sure about that? The T's cancel out.
fix it and show whatcha got, and we can go from there.
thats right. i always overthink things never even looked at the T's cancelling
To find the value of T for which the efficiency E is the greatest, we need to find the maximum point of the efficiency function.
To do that, we can take the derivative of the efficiency function with respect to T and set it equal to zero. Then we can solve for T.
Let's start by finding the derivative of the efficiency function:
E = T(1 - 0.35T) / (T + 0.35)
To simplify the equation, let's multiply through by the denominator (T + 0.35):
E(T + 0.35) = T(1 - 0.35T)
Expanding both sides:
ET + 0.35E = T - 0.35T^2
Now, let's differentiate both sides with respect to T:
dE/dT + 0.35(dE/dT) = 1 - 0.7T
Combining like terms:
1.35(dE/dT) = 1 - 0.7T
Dividing both sides by 1.35:
dE/dT = (1 - 0.7T) / 1.35
Now set the derivative equal to zero and solve for T:
(1 - 0.7T) / 1.35 = 0
Simplifying and multiplying through by 1.35:
1 - 0.7T = 0
-0.7T = -1
T = 1.43
So the efficiency is the greatest when T = 1.43, for T > 0.