Question

A trash 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.

Answer 1

Answer:

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 $z=\frac{1080}{xy}$ 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)$(\frac{1080}{xy})$

A = 5xy +$8(\frac{1080}{y})+8(\frac{1080}{x})$

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

$\frac{dA}{dx}=0$ and $\frac{dA}{dy}=0$

$\frac{dA}{dx}=5y+8(1080)(0)+8(1080)(-\frac{1}{y^{2}})=0$

5y =$\frac{8\times1080}{y^{2} }$

5y³ = 8640

y³ =$\frac{8640}{5}=1728$

y = 12 ft

For $\frac{dA}{dy}=0$

$\frac{dA}{dy}=5x+\frac{8(-1080)}{x^{2}}$=0

5x =$\frac{8(1080)}{x^{2} }$

5x³ = 8640

x³ = 1728

x = 12

Now from equation 1

z =$\frac{1080}{x}$

  =$\frac{1080}{144}$

z = 7.5

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