Question

Waste management company wants to construct a dumpster in the shape of a rectangular solid with no top whose length is four times its width. Its volume is fixed at 25. 6 yd3. Find the dimensions of the dumpster that will minimize its surface area.

Answer 1

To minimize the surface area of the dumpster, we need to find the dimensions of the rectangular solid that will give us the fixed volume of 25.6 yd^3.

Step-by-step explanation:

To minimize the surface area of the dumpster, we need to find the dimensions of the rectangular solid that will give us the fixed volume of 25.6 yd^3. Let's assume the width of the dumpster is x. Since the length is four times the width, the length will be 4x. And since the top is not included, the height will be x as well. To find the dimensions that minimize the surface area, we need to differentiate the surface area formula with respect to x, set it equal to zero, and solve for x. The formula for the surface area of the rectangular solid is 2lw + 2lh + 2wh. After differentiating and solving, we find that x = 1.6 yd, 4x = 6.4 yd, and the height is also 1.6 yd.

Answer 2

To minimize the surface area of a topless rectangular dumpster with a fixed volume of 25.6 yd3 and a length four times its width, we express the volume and surface area in terms of width, differentiate to find critical points, and then determine the optimal dimensions.

Step-by-step explanation:

Minimizing Surface Area of Rectangular Dumpster

To find the dimensions of a dumpster that minimize surface area, we can use calculus and the method of optimization. Given that the volume is fixed at 25.6 yd3 and the length is four times the width, we can express the volume as V = lwh. Since the length l is four times the width w, we have V = 4w2h. To find the surface area A to be minimized, we use the surface area formula for a rectangular prism with no top: A = lw + 2lh + 2wh, which simplifies to A = 4w2 + 8wh + 2wh using the relationship between l and w.

Using the fixed volume of 25.6 yd3, we can solve for h in terms of w: h = 25.6 / (4w2). Substituting this expression back into the surface area formula, we get A solely in terms of w which can be differentiated and set to zero to find the minimum surface area. Through the process of differentiation and finding critical points, we can find the optimal width w, then use it to solve for the optimal height h. Calculating the dimensions that minimize surface area will give us the most efficient design for the dumpster.

The surface area of the rectangular solid, or dumpster, can be minimized by calculating the optimal width using the given fixed volume and the resulting height, considering the length is four times the width.