Ask Question
7 June, 17:15

A trash company is designing an open-top, rectangular container that will have a volume of 1715 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.

+5
Answers (1)
  1. 7 June, 19:59
    0
    14 ft * 14 ft * 8.75 ft

    Step-by-step explanation:

    A garbage company is designing an open rectangular container that should have a volume of 1,715 cubic feet.

    So we have the length of the container = "x" ft, the width of the container = "y" ft and the height of the container = "z" ft

    Therefore the volume of the rectangular container would be:

    x * y * z = 1715 ft³

    z = 1715 / x * y

    The cost of making the bottom of the container is $ 5 per square foot, that is:

    5 * (x * y)

    Now, area of all sides of the container would be:

    2 * (x * z + y * z) = 2 * z * (x + y)

    We know that it has been given that the cost of making all the sides of the container is = $ 4 per square foot, so:

    4 * (2 * z * (x + y)) = 8 * z * (x + y)

    In total the costs would be:

    5 * (x * y) + 8 * z * (x + y)

    If we replace z, in the previous equation we have:

    5 * (x * y) + 8 * (1715 / x * y) * (x + y)

    solving, and we would have that the total cost would be:

    C = 5 * (x * y) + 13720 / x + 13720 / y

    Now we will find the derivative of C and make it equal to zero:

    dC / dx = 0; dC / dy = 0

    For dC / dx = 0:

    dC / dx = 5 * y + 13720 * - 1 / (y ^ 2) + 13720 * 0

    0 = 5 * y - 13720 / y ^ 2

    5 * y = 13720 / y ^ 2

    y ^ 3 = 13720/5 = 2744

    y = 14

    For dC / dy = 0:

    dC / dy = 5 * x + 13720 * 0 + 13720 * - 1 / (x ^ 2)

    0 = 5 * x - 13720 / x ^ 2

    5 * x = 13720 / x ^ 2

    x ^ 3 = 13720/5 = 2744

    x = 14

    now for z:

    z = 1715 / (14 * 14)

    z = 8.75

    Therefore, the dimensions of the container should be 14 ft * 14 ft * 8.75 ft to minimize manufacturing cost.
Know the Answer?
Not Sure About the Answer?
Find an answer to your question ✅ “A trash company is designing an open-top, rectangular container that will have a volume of 1715 ft cubed. The cost of making the bottom of ...” in 📘 Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions.
Search for Other Answers