Question

A trucking company must deliver a product to a location 150 miles away. The company must pay the driver a wage of $14 per hour. The company must pay for fuel, which costs C=v^2/250 dollars per hour, where v is the speed of the truck in miles per hour. How fast should the company ask the driver to travel in order to minimize the company's costs?

Answer 1

Speed of the truck should be 64.03 miles per hour to minimize the cost.

Step-by-step explanation:

Data provided in the question:

Distance = 150 miles

Wage = $14 per hour

Cost of fuel = ( v² ÷ 250 )

Now,

Total time taken = Distance ÷ speed

= 150 ÷ v

Therefore,

Total cost, TC = Wage + Cost of fuel

= $14 × (150 ÷ v) + ( v² ÷ 250 )

=
`(2100)/(v)+(v^2)/(250)`

for point of minima differentiating with respect to 'v'

TC'(v) =
`-(2100)/(v^2)+(2v)/(250)` = 0

or

`-(2100)/(v^2)+(2v)/(250)` = 0

or

`(v)/(125)=(2100)/(v^2)`

or

v³ = 2100 × 125

or

v = ∛262500

or

v = 64.03 miles per hour

hence,

Speed of the truck should be 64.03 miles per hour to minimize the cost.