Problem 39 - Contemporary Precalculus

Question

Find the dimensions of a box with a square base that has a volume of 867 cubic inches and the smallest possible surface area, as follows. (a) Write an equation for the surface area \(S\) of the box in terms of \(x\) and \(h .[\) Be sure to include all four sides, the top, and the bottom of the box.] (b) Write an equation in \(x\) and \(h\) that expresses the fact that the volume of the box is 867 . (c) Write an equation that expresses \(S\) as a function of \(x\). [Hint: Solve the equation in part (b) for \(h\), and substitute the result in the equation of part (a).] (d) Graph the function in part (c), and find the value of \(x\) that produces the smallest possible value of \(S .\) What is \(h\) in this case?

Step-by-Step Solution

Answer

Answer: The approximate dimensions of the box are x = 9 inches and h ≈ 10.69 inches.

Step 1: Write surface area equation

The surface area \(S\) of a box with a square base and dimensions \(x\) and \(h\) can be written as: $$S=2x^2 + 4xh$$ Here, \(2x^2\) is the area of the top and bottom of the box, and \(4xh\) is the combined area of all four sides.

Step 2: Write volume equation

The volume, V, of a box with dimensions x and h is equal to the product of its dimensions. In our case, the volume is 867 cubic inches, so we can write the equation as: $$V = x^2h = 867$$

Step 3: Express S as a function of x

Now, we will solve the volume equation for h and substitute the result in the surface area equation: $$ h=\frac{867}{x^2} $$ Substituting the value of h in the surface area equation: $$ S(x) = 2x^2 + 4x\frac{867}{x^2} $$

Step 4: Graph S(x) and find the smallest value of S

Graph the function \(S(x)\) to find the value of \(x\) which produces the smallest possible surface area. Using a graphing calculator or software, you'll find the minimum value when \(x \approx 9\). Now, find h by substituting this value of x back into the equation for h found in step 3: $$ h = \frac{867}{9^2} \approx 10.69 $$ The dimensions of the box with a square base, volume 867 cubic inches, and the smallest possible surface area are approximately \(x=9\) inches and \(h \approx 10.69\) inches.

Key Concepts

Surface Area of a Box

Understanding the surface area of a box is crucial for a variety of real-world applications such as packaging, material economy, and construction. Simply put, the surface area is the total area that the surface of the object occupies.

For a box with a square base, the surface area, denoted as \( S \), can be calculated by adding the areas of all six faces. Since the base is square, two faces are square with an area of \( x^2 \) each, where \( x \) is the side of the square base. The remaining four faces are rectangular and are determined by the base side \( x \) and the height of the box \( h \). Therefore, the area of each of these faces is \( xh \), resulting in an equation for the total surface area:

\[ S = 2x^2 + 4xh \]
While solving problems, it's imperative to keep the units consistent to avoid errors when calculating areas, which are always in square units—like square inches in this instance.

Volume of a Box

Volume is a measure of the space enclosed by a 3-dimensional object, such as a box. It is often calculated to determine the capacity or the amount of space available inside the object. The volume of a box with a square base is given by multiplying the area of the base by the height of the box.

In mathematical terms, if a box has a base side of \( x \) and a height \( h \), the volume \( V \) can be expressed as:

\[ V = x^2h \]
For the given problem, the volume of the box is known to be 867 cubic inches. This gives us a crucial equation to work with and facilitates further steps in the problem involving the exploration of the relationship between the surface area and volume to minimize material usage or maximize storage capacity.

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in a variety of settings in precalculus. They are polynomial equations of the second degree, which means the highest exponent of the variable, typically \( x \), is 2. The general form of a quadratic equation is:

\[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.

Quadratic equations can describe parabolic shapes when graphed, and they have a wide range of applications like calculating areas, trajectories, and optimizing functions. In optimization problems, finding the minimum or maximum value of a quadratic function often involves completing the square or using the quadratic formula. In our box example, the surface area function in terms of \( x \) transforms into a quadratic equation. The objective is to minimize this equation to find the box dimensions that yield the smallest surface area possible, which aligns with the efficiency needs of material design and packaging.