A rectangular box with a square base is to be constructed from a sheet of cardboard that measures 12 inches by 15 inches. The box has an open top and is to have a volume of 100 cubic inches.
a) Determine the dimensions of the box that minimize the amount of cardboard used.
b) What is the minimum amount of cardboard used?
a) Let's start by assuming the length of the side of the square base to be 'x' inches.
Since the box has a square base, the width of the box is also equal to 'x'.
The height of the box will be 'h' inches.
Given that the volume of the box should be 100 cubic inches, we can write the equation:
Volume of the box = length * width * height = x * x * h = 100
To minimize the amount of cardboard used, we need to find the dimensions that minimize the surface area of the box.
The surface area of the box consists of the area of the base and the four sides.
Area of the base = length * width = x * x = x^2
Area of the four sides = 2 * (length * height) + 2 * (width * height) = 2 * (x * h) + 2 * (x * h) = 4 * (x * h)
Total surface area of the box = Area of the base + Area of the four sides = x^2 + 4 * (x * h)
Now, we need to write the surface area equation in terms of a single variable to apply calculus.
Since we know that the length of the sheet of cardboard is 12 inches, and the width is 15 inches, we can write:
2 * (length * width) + 2 * (length * height) + 2 * (width * height) = 2 * (12 * 15) + 2 * (12 * h) + 2 * (15 * h) = 360 + 24h + 30h = 360 + 54h
Now, the surface area equation in terms of 'x' becomes:
Surface area = x^2 + 4 * (x * h) = x^2 + 4xh
To eliminate 'h' from the equation, we can substitute the value of 'h' found from the volume equation:
x^2 + 4xh = x^2 + 4x(100 / (x^2))
Simplifying the equation:
Surface area = x^2 + 400 / x
Now, we have the expression for the surface area of the box in terms of 'x' only.
Next, we need to find the critical points by taking the derivative of the surface area equation and setting it equal to zero.
d(Surface area)/dx = 2x - 400 / x^2 = 0
Multiplying both sides by x^2:
2x^3 - 400 = 0
2x^3 = 400
Dividing both sides by 2:
x^3 = 200
Taking the cube root:
x = ∛200
Using a calculator:
x ≈ 5.848
Since we're dealing with dimensions, we discard negative values and keep the positive value.
Therefore, the dimensions of the square base are approximately 5.848 inches by 5.848 inches.
b) To find the minimum amount of cardboard used, we substitute the value of 'x' into the surface area equation:
Surface area = x^2 + 400 / x
Surface area = (5.848)^2 + 400 / (5.848)
Surface area ≈ 34.13 + 68.42
Surface area ≈ 102.55
Therefore, the minimum amount of cardboard used is approximately 102.55 square inches.
Answer:
a) The dimensions of the box that minimize the amount of cardboard used are approximately 5.848 inches by 5.848 inches.
b) The minimum amount of cardboard used is approximately 102.55 square inches.