Question

An open-top storage bin in the shape of rectangular prism whose base is a square is constructed such that the volume is 3500 cm3. To minimize the weight of the storage bin, one must minimize the surface area. Find the surface area (in cm2) that minimizes weight.

Answer 1

To minimize the weight of the storage bin, we need to minimize the surface area. By deriving the surface area equation and setting the derivative equal to zero, we find the value of x that minimizes surface area. The surface area that minimizes weight is approximately 3236.37 cm².

Step-by-step explanation:

To minimize the weight of the storage bin, we need to minimize the surface area. Since the base is a square, let's assume the side length of the base is x.

The volume of the storage bin is given as 3500 cm³, which is equal to the product of the base area and height:
V = x² * h = 3500 cm³

To minimize the surface area, we need to find the value of x that minimizes the perimeter of the base and the area of the four side faces.
The surface area, A, of the storage bin is given by:
A = x² + 4xh

Using the volume equation, we can rearrange it to express h in terms of x:
h = 3500 / x²

Substituting this into the surface area equation, we get:
A = x² + 4x(3500 / x²)
Simplifying, we have:
A = x² + 14000 / x

To find the surface area that minimizes weight, we need to find the minimum value of A. We can do this by taking the derivative with respect to x and setting it equal to zero:

dA/dx = 2x - 14000/x² = 0

Multiplying through by x², we have:
2x³ - 14000 = 0
2x³ = 14000
x³ = 7000
x = ∛(7000) ≈ 19.19

Since x represents the side length of the base, we round it to the nearest whole number to get a practical value:
x ≈ 19 cm

The surface area that minimizes weight is then given by substituting this x value into the surface area equation:
A ≈ (19)² + 4(19)(3500/19²) ≈ 289 + 2947.37 ≈ 3236.37 cm²

Answer 2

The surface area that minimizes weight is equal to 1381.45 cm².

In Mathematics and Geometry, the surface area of a rectangular prism can be calculated and determined by using this mathematical equation or formula:

SA = 2(lh + lw + wh)

Where:

SA represents the surface area of a rectangular prism.
l represents the length of a rectangular prism.
w represents the width of a rectangular prism.
h represents the height of a rectangular prism.
Since the base of this rectangular prism is a square, the surface area function is given by;

SA = 2l² + 2lh + 2lh

SA = 2l² + 4lh.

For the volume function, we have:

V = lwh

3500 = l²h

h = 3500/l²

Next, we would substitute the value of h into the surface area function:

SA = 2l² + 4l(3500/l²)

SA = 2l² + 14000/l

In order to determine the minimum value of the surface area function, we would take the first derivative and set it equal to zero:

dSA/dl = 4l - 14000/l²

4l - 14000/l² = 0

4l³ - 14000 = 0

l³ = 3500

l = 15.18 cm.

For the value of h, we have:

h = 3500/15.18²

h = 15.18 cm

Now, we can determine the minimum surface area:

SA = 2(15.18)² + 4(15.18)(15.18)

SA = 460.52 + 920.93

SA = 1381.45 cm².