1. ## [SOLVED] minimums

A traveller employs a man to drive him from sydney to Wollongong for a hourly payment of P dollars. Running costs of the car, which are also paid by the traveller are kv^3 dollars per hour, where vkn/h is the speed and k is a constant. Find the uniform speed that will minimize the total cost.

i don't get it T_T

2. Hello, chibiusagi!

We're having trouble reading what you wrote.
Let me take a guess at what you meant . . .

A traveller employs a driver from Sydney to Wollongong at $\displaystyle P$ dollars per hour.
Operating cost of the car is $\displaystyle kv^3$ dollars per hour,
where $\displaystyle v$ km/hr is the speed and $\displaystyle k$ is a constant.
Find the uniform speed that will minimize the total cost.

Let $\displaystyle H$ = number of hours of driving.

The driver is paid $\displaystyle P$ dollars per hour for $\displaystyle H$ hours.
. . This will cost: .$\displaystyle PH$ dollars.

The car will cost $\displaystyle kv^3$ dollars per hour for $\displaystyle H$ hours.
. . This will cost: .$\displaystyle kv^3H$ dollars.

The total cost is: .$\displaystyle C \;=\;PH + kv^3H$ dollars. .[1]

Let $\displaystyle D$ = distance from Sydney to Wollongong (km).
. . At $\displaystyle v$ km/hr, it will take: .$\displaystyle H \:=\:\frac{D}{v}$ hours. .[2]

Substitute [2] into [1]: .$\displaystyle C \;=\;kv^3\left(\frac{D}{v}\right) + P\left(\frac{D}{v}\right) \;=\;kDv^2 + PDv^{-1}$

Minimize: .$\displaystyle C' \;=\;2kDv - PDv^{-2} \:=\:0$

Multiply by $\displaystyle v^2\!:\;\;2kDv^3 - PD \:=\:0 \quad\Rightarrow\quad 2kDv^3 \:=\:PD\quad\Rightarrow\quad v^3 \:=\:\frac{P}{2k}$

Therefore: .$\displaystyle v \;=\;\sqrt[3]{\frac{P}{2k}}$