13. Comp-Tech is a new firm that designs a rectangular compost, container that will be twice as tall as it is wide and must hold $18 \mathrm{ft}^{3}$ of composted food scraps. find the dimensions of the compost container with minimal surface area (include the top and bottom ). $1.55 \mathrm{ft}$ by $2.16 \mathrm{ft}$ by $3.14 \mathrm{ft}$ $2.84 \mathrm{ft}$ by $1.78 \mathrm{ft}$ by $3.56 \mathrm{ft}$ $2.13 \mathrm{ft}$ by $1.16 \mathrm{ft}$ by $3.08 \mathrm{ft}$ $2.19 \mathrm{ft}$ by $1.25 \mathrm{ft}$ by $3.02 \mathrm{ft}$
Solution
Step 1 ：The problem is asking for the dimensions of a rectangular compost container that has a minimal surface area. The container is twice as tall as it is wide and must hold 18 cubic feet of composted food scraps.
Step 2 ：We can start by setting up the equations for the volume and surface area of the rectangular compost container. The volume V of a rectangular prism is given by the formula \(V = lwh\), where l is the length, w is the width, and h is the height. The surface area A of a rectangular prism is given by the formula \(A = 2lw + 2lh + 2wh\).
Step 3 ：Given that the height is twice the width, we can substitute \(h = 2w\) into the volume equation to get \(V = l(2w)w = 2lw^2\). Solving for l, we get \(l = V / (2w^2)\).
Step 4 ：We can then substitute \(l = V / (2w^2)\) into the surface area equation to get \(A = 2(V / (2w^2))w + 2(V / (2w^2))(2w) + 2w(2w) = V/w + 2V/w + 4w^2\).
Step 5 ：To find the width w that minimizes the surface area, we can take the derivative of A with respect to w, set it equal to zero, and solve for w.
Step 6 ：The calculated dimensions of the compost container that minimize the surface area are approximately 1.44 ft for the width, 2.88 ft for the length, and 2.88 ft for the height. However, these values are not exactly the same as any of the options given in the question.
Step 7 ：The rounded dimensions of the compost container that minimize the surface area are approximately 1.89 ft for the width, 2.52 ft for the length, and 3.78 ft for the height. These values do not match any of the options given in the question.
Step 8 ：Final Answer: The dimensions of the compost container with minimal surface area are approximately \(\boxed{1.89 \mathrm{ft}}\) by \(\boxed{2.52 \mathrm{ft}}\) by \(\boxed{3.78 \mathrm{ft}}\).
