Question

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

#Answer# The relation between h and x for the given volume is: h = 125/x². The surface area function S(x) is: S(x) = x² + 500/x. Graphing S(x), we can estimate that the minimum surface area occurs for x approximately between 5 and 6. By observing the graph, we can determine the approximate minimal surface area and its corresponding x value accordingly.

Step-by-step solution

Find the relation between \(h\) and \(x\).

We know that the volume \(V\) of the box is given by the product of the length, width, and height, and that \(V = 125 \, ft^3\). Since the base is square, the volume can be expressed as $ V = x^2h. $ We can find the relation between \(h\) and \(x\) by setting \(V = 125\) and solving for \(h\): $ h = \frac{V}{x^2} = \frac{125}{x^2}. $ Step 2: Compute the surface area function S(x)

Compute \(S(x)\)

The surface area \(S\) of the lidless box can be computed as the sum of the 4 identical side faces and the base. The surface area \(S(x)\) can be expressed as $ S(x) = x^2 + 4(xh). $ Now substitute the expression for \(h\) found in Step 1: $ S(x) = x^2 + 4\left(x\frac{125}{x^2}\right) = x^2 + \frac{500}{x}. $ Step 3: Graph the function S(x)

Graph \(S(x)\)

To graph the function \(S(x) = x^2 + \frac{500}{x}\), consider the domain of the function to be all positive numbers (\(x > 0\)). You can use a graphing calculator or a software tool like Desmos to plot this function. Make sure to label the graph and the axes. Step 4: Estimate the minimum surface area value

Estimate the minimum surface area value

Based on the graph of \(S(x) = x^2 + \frac{500}{x}\), you can visually estimate the value of \(x\) that results in the minimal surface area. The minimum surface area can be observed when the graph has its lowest point. By looking at the graph, we can estimate this \(x\) value and its corresponding minimal surface area.

Key concepts

Calculus

Let's explore how calculus helps us solve optimization problems, like minimizing the surface area of a box. In calculus, finding the minimum or maximum values of a function involves understanding the function's first and second derivatives. These are powerful tools that allow us to determine critical points, which are potential candidates for these extremes.
This exercise is about finding the minimum surface area of a box under a given volume constraint. To approach this task, we can use the derivative of the surface area function, \(S(x)\). By setting the derivative to zero, we find critical points that could represent either minimum or maximum values.

• First, calculate the derivative of the surface area function \(S(x)\).
• Next, solve the equation resulting from setting the derivative equal to zero to find critical points.
• Determine if these points correspond to a minimum surface area using the second derivative test.

This process not only finds the solution to the problem but also demonstrates the practical applications of calculus.

Graphing Functions

Graphing functions is a visual method to analyze and understand the behavior of equations. In this problem, we graph the surface area function \(S(x) = x^2 + \frac{500}{x}\). This graph helps us visually identify the point where the surface area is minimized, balancing the trade-off between the size of the box base and height.
To graph this function, you can use a graphing tool or calculator. The steps are as follows:

• Plot the graph for values of \(x > 0\) since the box dimensions must be positive.
• Examine the shape of the curve – this function is a combination of a quadratic \(x^2\) and a rational function \(\frac{500}{x}\), leading to a curve that decreases sharply, reaches a minimum, and then increases.
• Identify the lowest point on the graph, which is the global minimum indicating the smallest surface area for the given volume.

Understanding graphing can make complex algebraic expressions more intuitive and accessible.

Volume of Box

Understanding the volume of a box is essential to solving this problem. The volume is a measure of the space inside the box, calculated by multiplying the base area by the height. Here, the base is a square, so the formula becomes simple: \(V = x^2h\). Given the volume constraint \(V = 125 \, \mathrm{ft}^{3}\), it helps in finding a relationship between \(x\) and \(h\).
Here’s a step-by-step breakdown:

• Set the volume \(V\) equal to \(x^2h\), then solve for \(h\) to find \(h = \frac{125}{x^2}\). This shows how height varies inversely with the square of the base side length.
• This relationship allows substitution into the surface area formula, bringing us to an equation solely in terms of \(x\), making it simpler to analyze and minimize.
• Applying constraints effectively to manipulate these formulas is a key step in problems involving geometry and optimization.

The concept of volume ensures that the box meets specific project criteria while optimizing other properties.