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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?

Respuesta :

Answer:

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

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 )

= [tex]\frac{2100}{v}+\frac{v^2}{250}[/tex]

for point of minima differentiating with respect to 'v'

TC'(v) =  [tex]-\frac{2100}{v^2}+\frac{2v}{250}[/tex] = 0

or

[tex]-\frac{2100}{v^2}+\frac{2v}{250}[/tex] = 0

or

[tex]\frac{v}{125}=\frac{2100}{v^2}[/tex]

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.

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