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 $P$ dollars per hour.
Operating cost of the car is $kv^3$ dollars per hour,
where $v$ km/hr is the speed and $k$ is a constant.
Find the uniform speed that will minimize the total cost.

Let $H$ = number of hours of driving.

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

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

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

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

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

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

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

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