Question

Optimal boxes Imagine a lidless box with height \(h\) and a square base whose sides have length \(x\). The box must have a volume of \(125 \mathrm{ft}^{3}\). a. Find and graph the function \(S(x)\) that gives the surface area of the box, for all values of \(x>0\) b. Based on your graph in part (a), estimate the value of \(x\) that produces the box with a minimum surface area.

Short answer

Question: Based on the given information and steps, estimate the value of x that produces the minimum surface area of the lidless box. Answer: x ≈ 3.7 ft.

Step-by-step solution

Find the relationship between height and side length

Since the box has a volume of 125 ft³ and a square base of side length x, we can write the volume equation as: \[V = x^2h\] We are given that V is 125 cubic feet, so we have: \[125 = x^2h\] Solve this equation for h to express it in terms of x: \[h = \frac{125}{x^2}\]

Find the surface area function S(x)

Now, we need to find the surface area function S(x) for the lidless box. The surface area S(x) is the sum of the areas of the base and four sides. Since it's a lidless box, we only need to find the area for the base and 4 sides. The base is a square with side length x, and its area is x^2. Each side has an area of xh. We have 4 sides, so the total surface area from the sides is 4xh. Combine the areas of the base and sides: \[S(x) = x^2 + 4xh\] Now, replace h with the expression we found in Step 1: \[S(x) = x^2 + 4x\left(\frac{125}{x^2}\right)\] Simplify the expression: \[S(x) = x^2 + \frac{500x}{x^2}\]

Graph the function S(x) for all values of x>0

Now, graph the function S(x) with x-axis representing the side length x, and the y-axis representing the surface area S(x), for all positive values of x. Since it's difficult to graph it here in text form, you can use a graphing calculator or online graphing tool to visualize the function.

Estimate the value of x for minimum surface area based on the graph

Observe the graph obtained in Step 3 to find the lowest point of the curve. This point corresponds to the minimum surface area of the box. Estimate the x-value of this lowest point. From the graph, we can observe that the minimum value of the surface area S(x) occurs at about x ≈ 3.7 ft.

Key concepts

Surface Area of a Lidless Box

When calculating the surface area of a lidless box with a square base, you need to consider both the base and the four sides.
The base of the box is a square with side length \(x\), so its area is simply \(x^2\). While the sides contribute to the surface area as well, remember this box does not have a lid.
Each of the four sides of the box is a rectangle with dimensions of \(x\) (the width) and \(h\) (the height). The area of each side is therefore \(xh\). But since there are four sides, you multiply the area of one side by four: \(4xh\).

• Base Area: \(x^2\)
• Total Side Area: \(4xh\)

Adding these areas together gives the total surface area function \(S(x)\):
\[S(x) = x^2 + 4xh\].
Plugging in for \(h\) using the volume equation \(h = \frac{125}{x^2}\), you can further simplify the expression for surface area.

Volume Equation of a Box

The volume of the given box must remain constant at 125 cubic feet. This constraint is vital for setting up the surface area function.
The volume of a box is calculated using the formula \(V = x^2h\), where \(x\) is the side length of the square base and \(h\) is the height. Since this volume is fixed, the equation becomes:
\[x^2h = 125\].
You can solve this equation to express the height \(h\) in terms of \(x\), isolating \(h\) on one side:
\(h = \frac{125}{x^2}\).
This step allows you to easily substitute \(h\) in the surface area function. By doing so, the function for the surface area becomes dependent solely on \(x\), which is crucial for the optimization of surface area.

Graphing Functions for Optimization

To find the optimal design of the box, graphing the surface area function \(S(x)\) helps visually identify the minimum point. This is where calculus optimization shines!
When graphing \(S(x) = x^2 + \frac{500}{x}\), you'll notice a curve on the xy-plane with the x-axis representing \(x\) and the y-axis showing the surface area. The graph lets you easily spot the lowest point indicating minimal surface area.

• Use graphing tools to plot the function accurately.
• Look for the vertex or the lowest part of the curve; this is your optimal point.
• Ensure x remains positive, as negative values for length do not make sense in this context.

From the graph, you can reasonably estimate where the minimum occurs, which here is around \(x \approx 3.7\). This insight is crucial in determining the right dimensions for the box to use the least material.