If an open box has a square base and a volume of 112 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (Round your answers to two decimal places.)
Height:
Length:
Width:
if side is x,
height is 112/x^2
area
a = x^2 + 4xh
= x^2 + 4x(112/x^2)
= x^2 + 28/x
da/dx = 2x - 28/x^2
da/dx=0 when x^3 = 14
and so on
To find the dimensions of the box with minimum material, we need to optimize the amount of material used. Since the box has a square base, let's assume the length of the base is "x".
The area of the base is the side length squared, so the volume of the box is given by the formula:
Volume = x^2 * height
We know that the volume is 112 in^3, so we can rearrange the formula to solve for the height:
height = Volume / (x^2)
Now, we need to find the minimum amount of material used. The surface area of the box can be calculated by adding the areas of the base and the four sides:
Surface Area = 2*(base area) + 4*(side area)
Since the base is a square, the base area is:
Base Area = x^2
The side area can be calculated by multiplying the length of the side by the height. Since the sides are rectangles, the length of the side is x and the height is the same as the height of the box:
Side Area = x * height = x * [Volume / (x^2)]
Now, we can substitute the value of height into the equation for the side area:
Side Area = x * [Volume / (x^2)] = Volume / x
Finally, we can calculate the surface area by substituting the values:
Surface Area = 2*(x^2) + 4*(Volume / x) = 2x^2 + 4(112/x)
To minimize the amount of material, we need to find the value of "x" that minimizes the surface area. We can do this by taking the derivative of the surface area equation with respect to "x", setting it to zero, and solving for "x".
Derivative of Surface Area with respect to "x":
d(Surface Area) / dx = 4x - 448/(x^2)
Setting the derivative equal to zero:
4x - 448/(x^2) = 0
Multiplying through by x^2:
4x^3 - 448 = 0
Simplifying:
x^3 = 112
Taking the cube root of both sides:
x = ∛112
Calculating the cube root of 112:
x ≈ 4.84
So, the length and width of the base of the box are approximately 4.84 inches.
To find the height, we can substitute this value back into the formula for the height:
height = Volume / (x^2) = 112 / (4.84^2)
Calculating this value:
height ≈ 4.84 inches
Therefore, the dimensions of the box with a minimum amount of material used are:
Height: 4.84 inches
Length: 4.84 inches
Width: 4.84 inches