1. A box with a square base and an open top must have a volume of 4,000 cm\(^3\). 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 m\(^3\). The length of the 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.

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

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

Answered by Joel0920 | 2024-06-10

The dimensions of the box with a volume of 4000 cm³ that minimizes the material used are 20 cm x 20 cm x 10 cm. The cost for constructing the least expensive storage container with a volume of 10 m³ is approximately $327.08. These results are achieved through using calculus to minimize surface area and cost under given constraints.
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Answered by Joel0920 | 2025-01-15