Atrash company is designing an open-top, rectangular container that will have a volume of 1080 ft cubed. the cost of making the bottom of the container is $5 per square foot, and the cost of the sides is $4 per square foot. find the dimensions of the container that will minimize total cost.

Dimensions of the container should be 12×12×7.5 ft to minimize the making cost.

Step-by-step explanation:

A trash company is designing an open top, rectangular container having volume = 1080 ft³

Let the length of container = x ft , width of the container = y ft and height of the container = z ft.

So volume of the rectangular container = xyz = 1080 ft³

Or ft -----(1)

Cost of making the bottom of the container = $5 per square ft

Area of the bottom = xy

Cost of making the bottom @ $5 per square ft = 5xy

Area of all sides of the container = 2(xz + yz) = 2z(x+ y)

Now it has been given that cost of making all sides of the container is = $4 per square ft

So total cost to manufacture sides = 4[2z(x + y)]

Now cost of making bottom and sides of the container = 5xy + 8z(x + y)

We put the value of z from equation 1

Total cost A = 5xy+8(x + y)

A = 5xy +

Now we will find the derivative of A and equate it to the zero

and

5y =

5y³ = 8640

y³ =

y = 12 ft

For

=0

5x =

5x³ = 8640

x³ = 1728

x = 12

Now from equation 1

z =

  =

z = 7.5

Therefore, dimensions of the container should be 12×12×7.5 ft to minimize the making cost.

b

step-by-step explanation:

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