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

To minimize the total cost, we need to minimize the cost of the bottom and the cost of the sides. We can find the dimensions of the container that will minimize the total cost by solving a system of equations.

Step-by-step explanation:

Let's assume that the length of the rectangular container is x ft, the width is y ft, and the height is z ft.

The volume of the container is given as 1080 ft3.

Therefore, we have the equation:

x * y * z = 1080

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.

The cost of the bottom is 5 * (x * y).

The cost of the sides is 4 * (2xy + 2xz + 2yz).

To minimize the total cost, we need to minimize the cost of the bottom and the cost of the sides.

First, let's solve the volume equation for x:

x = (1080) / (y * z)

Substituting the value of x in the cost equation, we have:

Cost = 5 * (1080) / (y * z) * y + 4 * (2 * (1080 / (y * z)) * y + 2 * (1080) / (y * z) * z + 2 * y * z)

Now, we can find the minimum cost by taking the derivative of the cost equation with respect to y and z, and setting it equal to zero.

Then, we solve the resulting system of equations to find the values of y and z that minimize the cost.

Finally, we substitute the values of y and z back into the volume equation to find the value of x.

By solving the equations, we can find the dimensions of the container that will minimize the total cost.

Answer 2

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=(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)
`((1080)/(xy))`

A = 5xy +
`8((1080)/(y))+8((1080)/(x))`

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

`(dA)/(dx)=0` and
`(dA)/(dy)=0`

`(dA)/(dx)=5y+8(1080)(0)+8(1080)(-(1)/(y^(2)))=0`

5y =
`(8*1080)/(y^(2) )`

5y³ = 8640

y³ =
`(8640)/(5)=1728`

y = 12 ft

For
`(dA)/(dy)=0`

`(dA)/(dy)=5x+(8(-1080))/(x^(2))`=0

5x =
`(8(1080))/(x^(2) )`

5x³ = 8640

x³ = 1728

x = 12

Now from equation 1

z =
`(1080)/(x)`

=
`(1080)/(144)`

z = 7.5

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