Problem 84 - College Algebra

Question

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question. An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. Let \(x=\) length of the side of the base.

Step-by-Step Solution

Answer

The dimensions are 6 inches for the base side and 3 inches for the height.

Step 1: Understand the Problem

We need to construct a rational function representing the surface area of an open box. The box has a square base of side length \(x\) and a volume of 108 cubic inches. Our goal is to find the dimensions that minimize the surface area.

Step 2: Establish the Relationship Between Variables

For a box with a square base, the volume \(V\) is given by \(V = x^2 h\), where \(h\) is the height of the box. Given \(V = 108\), we have:\[x^2 h = 108\]From this, we can express the height \(h\) in terms of \(x\):\[h = \frac{108}{x^2}\]

Step 3: Formulate the Surface Area Function

The surface area \(A\) of the open box (without a top) is the sum of the area of the base and the four sides. Therefore, it is given by:\[A = x^2 + 4xh\]Substituting \(h = \frac{108}{x^2}\) from the previous step:\[A = x^2 + 4x\left(\frac{108}{x^2}\right) = x^2 + \frac{432}{x}\]

Step 4: Differentiate to Find the Minimum

To find the minimum surface area, take the derivative of the surface area function \(A(x) = x^2 + \frac{432}{x}\) with respect to \(x\):\[A'(x) = 2x - \frac{432}{x^2}\]Set \(A'(x) = 0\) to find critical points:\[2x - \frac{432}{x^2} = 0\]Solve for \(x\) to find the length that minimizes the surface area.

Step 5: Solve the Equation

To solve \(2x - \frac{432}{x^2} = 0\), rearrange the equation:\[2x^3 = 432\]Divide both sides by 2:\[x^3 = 216\]Taking the cube root of both sides gives:\[x = 6\]

Step 6: Verify and Solve for Remaining Dimensions

With \(x = 6\), substitute back into the height formula:\[h = \frac{108}{x^2} = \frac{108}{36} = 3\]Thus, the box dimensions are:Length of base side: 6 inchesHeight: 3 inches

Step 7: Calculate Surface Area for Verification

Substitute \(x = 6\) and \(h = 3\) into the surface area formula:\[A = x^2 + 4xh = 6^2 + 4 \times 6 \times 3 = 36 + 72 = 108\]The surface area is indeed minimized at 108 square inches.

Key Concepts

Volume of a Box

The volume of a box is a fundamental concept in geometry, particularly when dealing with three-dimensional shapes. For an open box with a square base, the volume can be calculated using its side length and height. The formula for the volume of such a box is:\[ V = x^2 h \]Here, \(x\) represents the side length of the base, and \(h\) is the height of the box. This formula expresses how much space the box occupies. When we know the volume and one of the dimensions, we can rearrange this formula to solve for the unknown dimension.

In our example, we know the volume is 108 cubic inches and need to express the height \(h\) in terms of the side length \(x\). By rearranging the formula, we get:\[ h = \frac{108}{x^2} \]Understanding how to manipulate this equation helps in solving for dimensions that meet specific criteria, such as minimizing surface area.

Surface Area

The surface area of a box, specifically an open box, constitutes all the areas of its surfaces that enclose the volume. For a box with an open top and a square base, the surface area consists of the base and the four sides. The formula is:\[ A = x^2 + 4xh \]The term \(x^2\) accounts for the area of the square base, and the term \(4xh\) accounts for the four identical sides of the box.

Substituting \(h = \frac{108}{x^2}\) gives the surface area in terms of a single variable:\[ A = x^2 + \frac{432}{x} \]The balance of the base size and box height impacts the overall surface area. Our task is to optimize these dimensions to minimize this total surface area.

Critical Points

Critical points in calculus are values of a variable where a function's derivative is zero or undefined. These points often correspond to minimum, maximum, or inflection points on a graph of the function.

For minimizing the surface area of a box, we set the derivative of the surface area function equal to zero:\[ A'(x) = 2x - \frac{432}{x^2} = 0 \]This equation yields the critical values for \(x\), where the surface area could be at a minimum or maximum. Solving for \(x\) will reveal the box's dimensions that minimize its surface area. In our scenario, solving the equation finds that the critical point (and indeed the minimum surface area point) occurs when \(x = 6\).

Differentiation

Differentiation is a process in calculus used to determine the rate at which a function is changing at any given point. This technique is particularly useful for optimization problems, such as finding minimum or maximum values of a function.

In this exercise, we differentiate the function representing the surface area in order to locate its critical points. The derivative \(A'(x)\) is calculated as:\[ A'(x) = 2x - \frac{432}{x^2} \]Setting this derivative equal to zero helps identify potential points of minimum or maximum surface area. Solving this equation provides the dimensions of the box that minimize the surface area, showcasing how differentiation can serve a practical purpose in geometrical optimization problems. This is an essential tool in many areas of mathematics and physics.