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Does anyone know the answer (s) to this?

Jimmy Perdon

New member
May 3, 2021
4
A simplified model of the power P required to sustain the motion of an electric car at speed v experiencing nonzero drag can be modeled by the equation:

P = Av^2 + (B/v)
(where A and B are positive constants.)

(a) What speed vP minimizes power?
(b) What power does the speed in (a) require?
(c) Suppose that an electric car has a usable store E of energy. How far dP can the electric car travel at the
speed found in (b)?
 

Greg

Perseverance
Staff member
Feb 5, 2013
1,404
Here's the derivative for the function of power, \(\displaystyle P=Av^2+\frac{B}{v} (\text{Where }A\text{ and }B\text{ are positive constants)} \) that they give in the introduction to the problem:

\(\displaystyle P'(v)=2Av-\frac{B}{v^2} \)

To answer \(\displaystyle \text{(a) What speed }vP\text{ minimizes power?} \) we set this expression equal to \(\displaystyle 0 \) and solve for \(\displaystyle v \):

\(\displaystyle
\begin{align*}
2Av-\frac{B}{v^2}&=0 \\
2Av&=\frac{B}{v^2} \\
2Av\cdot\frac{v^2}{B}&=\frac{B}{v^2}\cdot\frac{v^2}{B} \\
\frac{2Av\cdot v^2}{B}&=1 \\
\frac{2Av^3}{B}\cdot\frac{B}{2A}&=1\cdot\frac{B}{2A} \\
v^3&=\frac{B}{2A} \\
v=\sqrt[3]{\frac{B}{2A}} \\
\end{align*}
\)

Now we need to determine if this result is a minimum or a maximum and we do this by examining the concavity of the graph of \(\displaystyle P \)

I'll let you take a look at the graph and experiment with the different values for \(\displaystyle A \) and \(\displaystyle B \). As a suggestion, take a look at the graph when \(\displaystyle A \) and \(\displaystyle B \) are different signs.
 
Last edited:

skeeter

Well-known member
MHB Math Helper
Mar 1, 2012
938