Question

The power, in amps, transmitted by a belt drive from a motor is given by the function, P = 100v - (3/16)v^3, where v is the linear velocity of the belt, in meters per second.

a) For what value of v is the power at a maximum value?
b) What is the maximum power?

Answer 1

If we consider Power as a function of Velocity, then:

P = 100v - 3/16(v^3)

a) To find the value of (V) at maximum power, we need to take the derivative of P with respect to V, and set that derivative to zero:

P = 100v - 3/16(v^3)

then by the power rule:

dP/dV = 100 - 9/16(v^2)

we need to set dP/dV = 0, so plugging that in we get:

0 = 100 - 9/16 (v^2)

-100 = -9/16(v^2)

100 = 9/16(v^2)

1600/9 = v^2

v = +40/3, -40/3

Since we are looking for Maximum Power, the +40/3 will give us the highest value for Power output, so v = +40/3 is the velocity at which the belt will generate the maximum power.

b) To find the maximum power, we just need to plug in our v = 40/3 into the original equation and find the resulting P.

P = 100(v) - 3/16(v^3)

P = 100(40/3) - 3/16[(40/3)^3]

P = 4000/3 - 3/16 (64000/27)

P = 4000/3 - 4000/9

P = 12000/9 - 4000/9

P = 8000/9 amps or about 888.88... amps

Answer 2

WOW! THANKS SOO MUCH!! I did guess and check and got 13, but I had no idea to how to actually get the right answer. THANKS SO MUCH AGAIN! :)

Answer 3

glad to have helped haha