A rectangular box with square base (which is NOT necessarily a cube) and NO top is to be made to contain 9 cubic feet. The material for the base costs $2 per square foot and the material for the sides $3 per square foot. Find the dimensions that minimize the cost of the box AND �find the minimum cost of the box.

Answer 1

let the base be x by x ft
let the height be h ft

V = x^2 h
h = 9/x^2

Cost = 2(x^2) + 3(4xh)
= 2x^2 + 12x(9/x^2)
= 2x^2 + 108/x
dCost/dx = 4x - 108/x^2
= 0 for a min of Cost
4x = 108/x^2
x^3 = 27
x = 3
then h = 9/3^2 = 1

The min cost is when the base is 3 ft by 3 ft and the height is 1 foot
the minimum cost is 2x^2 + 108/x
= 18 + 108/9 = $30

Answer 2

Thanks for the help Reiny. Was wondering if you know how to setup the question bellow. Coz this is the one i find the hardest to do and had tried but don't get it?

A truck driver, on assignment from the owner of the truck is to drive on a 300 mile stretch of highway at a constant speed of v miles per hour. According to road signs, the minimum speed allowed is 55 miles per hour and the speed limit is 70 miles per hour. The cost of gas on the day of the trip is $2.60 per gallon and the gas in the truck has been measured to consumed at a rate of (1+(1/400)*(v^2) gallons per hour.

If the truck driver earns $20 per hour what speed v should the truck driver be assigned to drive in order to keep the cost for the owner of the company as low as possible? Find the minimum cost and compare it to the cost when the driver drives at 55 and 70 miles per hour.