Ask Question
15 February, 11:49

1. A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of material used.

2. A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $20 per square meter. Material for the sides costs $12 per square meter. Find the cost of materials for the cheapest such container.

+1
Answers (1)
  1. 15 February, 11:53
    0
    1. Let side of square base of the box = x cm

    and height of the box = y cm.

    Volume = x²y

    4000 = x²y

    y = 4000/x² ... (1)

    The material used for the box Surface Area = Areaof base + 4 times area of sides

    Surface Area, A = x² + 4xy

    Plug in value from (1) to get

    A = x² + 4x (4000/x²)

    A = x² + 16000/x

    To find optimal value, find derivative and equate to 0

    A' = 2x - 16000/x²

    0 = 2x - 16000/x²

    16000/x² = 2x

    x³ = 8000

    x = 20 cm

    y = 4000 / (20) ² = 10

    Dimension of the box is 20 cm x 20 cm x 10 cm

    2. Let width of the base of the box = x meter

    then length of the base = 2x meter

    and height of the box = y meter

    Volume of the box = x (2x) y = 2x²y

    10 = 2x²y

    y = 5/x² ... (1)

    Cost of box material, C = cost of base + cost of sides

    C = 20 (2x²) + 12 (2xy+4xy)

    C = 40x² + 72xy

    C = 40x² + 72x (5/x²) ... from (1)

    C = 40x² + 360/x

    C' = 80x - 360/x²

    0 = 80x - 360/x²

    360/x² = 80x

    x³ = 4.5

    x = 1.65 m

    y = 5 / (1.65) ² = 1.84 m

    Cost, C = 40x² + 360/x = 40 (1.65) ² + 360/1.65 = $327.08
Know the Answer?
Not Sure About the Answer?
Find an answer to your question 👍 “1. A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of ...” in 📗 Mathematics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers