Final answer:
To minimize the weight of the storage bin, we need to minimize the surface area. We can do this by finding the dimensions of the rectangular prism that will result in the smallest surface area. Using the volume and the equation for surface area, we can determine the dimensions and calculate the surface area that minimizes the weight.
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 its side length is s cm. The height of the rectangular prism can then be found by dividing the volume (3500 cm³) by the area of the base (s² cm²) to get a height of 3500/s cm. The surface area of the bin is given by 2s² + 4sh, where h is the height of the bin. Substituting the value of h, we have:
Surface Area (SA) = 2s² + 4s(3500/s)
Simplifying, we get:
SA = 2s² + 14000/s
To minimize the surface area, we can differentiate this equation with respect to s and equate it to zero:
d(SA)/ds = 4s - 14000/s² = 0
Solving this equation, we find s = 10 cm. Substituting this value back into the equation for SA, we get:
SA = 2(10)² + 14000/10 = 200 cm² + 1400 cm² = 1600 cm²