A rectangular box with a volume of $540 \mathrm{ft}^{3}$ is to be constructed with a square base and top. The cost per square foot for the bottom is $20 \notin$, for the top is $15 \%$, and for the sides is $2.5 \%$. What dimensions will minimize the cost?
What are the dimensions of the box?
The length of one side of the base is The height of the box is
(Round to one decimal place as needed.)

Step 1 ：The problem is asking for the dimensions of a box that will minimize the cost of construction given the cost per square foot of each side. The box has a square base and top, and a fixed volume.

Step 2 ：We can start by expressing the cost of the box in terms of one variable. The volume of the box is given by the formula \(V = lwh\), where \(l\) is the length, \(w\) is the width, and \(h\) is the height. Since the base and top are square, \(l = w\). So, we can express the volume as \(V = l^2h\).

Step 3 ：We can express \(h\) in terms of \(l\) and \(V\) as \(h = \frac{V}{l^2}\).

Step 4 ：The cost of the box is given by the formula \(C = 20l^2 + 15l^2 + 2.5(4lh)\), where the first term represents the cost of the bottom, the second term represents the cost of the top, and the third term represents the cost of the sides.

Step 5 ：We can substitute \(h = \frac{V}{l^2}\) into the cost formula to express \(C\) in terms of \(l\) only.

Step 6 ：Then, we can find the derivative of \(C\) with respect to \(l\) and set it equal to zero to find the value of \(l\) that minimizes \(C\).

Step 7 ：Finally, we can substitute this value of \(l\) back into the formula for \(h\) to find the height of the box.

Step 8 ：Final Answer: The dimensions that will minimize the cost are a base side length of \(\boxed{4.3}\) feet and a height of \(\boxed{29.8}\) feet.