Problem 61 - Thomas Calculus

Question

You are to construct an open rectangular box with a square base and a volume of 48 \(\mathrm{ft}^{3} .\) If material for the bottom costs \(\ 6 / \mathrm{ft}^{2}\) and material for the sides costs \(\ 4 / \mathrm{ft}^{2},\) what dimensions will result in the least expensive box? What is the minimum cost?

Step-by-Step Solution

Answer

Dimensions: 4 ft x 4 ft x 3 ft. Minimum cost: $288.

Step 1: Define the Variables

Let the side length of the square base be \( x \) feet and the height of the box be \( h \) feet. The volume of the box is given by \( x^2 h = 48 \ \, \mathrm{ft}^3 \).

Step 2: Express the Height in Terms of x

To express \( h \), use the volume equation: \[ h = \frac{48}{x^2} \]

Step 3: Calculate the Surface Area for Cost Calculation

The cost depends on the surface areas: the base \( x^2 \) and four sides each with area \( xh \). Therefore, the total area is \( x^2 + 4xh \).

Step 4: Express the Cost Function in Terms of x

The cost for the base is \( 6x^2 \) and the sides are \( 4 \times 4xh = 16xh \). Substitute \( h = \frac{48}{x^2} \) from Step 2: \[ C(x) = 6x^2 + 16x \left(\frac{48}{x^2}\right) = 6x^2 + \frac{768}{x} \]

Step 5: Differentiate the Cost Function

To minimize the cost, find the derivative of \( C(x) \): \[ C'(x) = 12x - \frac{768}{x^2} \]

Step 6: Find Critical Points

Set \( C'(x) = 0 \) to find critical points: \[ 12x - \frac{768}{x^2} = 0 \] Solving \( 12x^3 = 768 \) gives \( x^3 = 64 \), so \( x = 4 \) feet.

Step 7: Calculate Corresponding Height

Using \( x = 4 \), calculate \( h = \frac{48}{x^2} = \frac{48}{16} = 3 \) feet.

Step 8: Determine Minimum Cost

Substitute \( x = 4 \) and \( h = 3 \) into the cost function: \[ C(4) = 6(4^2) + \frac{768}{4} = 96 + 192 = 288 \]. The minimum cost is $288.

Key Concepts

Volume of a Box

When designing or optimizing boxes, understanding the concept of volume is crucial. The volume of a box, especially an open rectangular one with a square base, can be calculated using the formula:\[ V = x^2h \]where:- \( x \) is the side length of the square base- \( h \) is the height of the boxIn our case, the volume is set to a specific value of 48 cubic feet. This relationship means that if you adjust the base size \( x \), you must also adjust the height \( h \) to keep the volume constant. By rearranging the volume formula, you can express the height in terms of the base side length:\[ h = \frac{48}{x^2} \]This step is essential in determining other measurements and costs associated with constructing the box.

Cost Function

The cost function is an equation that represents how much it will cost to build something based on its dimensions. For the box, different materials incur different costs. Specifically, the base material costs \( 6 \, /\mathrm{ft}^2 \) and the sides cost \( 4 \, /\mathrm{ft}^2 \).To be able to manage and minimize the expenses effectively, you need to create a cost function that helps you understand and compute the total cost:\[ C(x) = 6x^2 + 16x \left(\frac{48}{x^2}\right) \]Each term of the function represents:- \( 6x^2 \): Cost of the base- \( 16x \left(\frac{48}{x^2}\right) \): Cost of the sidesThe goal in optimization is to make this cost function as small as possible, which is applicable for real-life resource and expense management.

Surface Area Calculation

Calculating the surface area is key when you want to assess the materials needed for construction and their respective costs. The surface area for our box comes from the base and the four sides:- Base Area: \( x^2 \)- Side Area: Each side has an area of \( xh \). With four sides, this is \( 4xh \).Therefore, the total surface area \( S \) involved in the cost calculation is:\[ S = x^2 + 4xh \]Substituting \( h \) from the volume formula, we relate surface area back to a single variable (\( x \)) to make further calculations feasible. This simplification is imperative for setting up the cost function and further derivative calculations.

Differentiation

Differentiation plays a central role in optimization problems. To minimize the material cost of our box, we need to examine how the cost changes with respect to the dimensions, particularly the base dimension \( x \).The derivative of the cost function \( C(x) \) with respect to \( x \) is:\[ C'(x) = 12x - \frac{768}{x^2} \]By setting this derivative to zero:\[ 12x - \frac{768}{x^2} = 0 \]we find the critical points, which could correspond to the minimum cost. Differentiation thus allows us to identify where changes in \( x \) result in cost reductions. It's a powerful tool in calculus for finding optimal solutions.

Critical Points

Critical points, found by setting the derivative of a function to zero, tell us where potential maxima or minima occur. In the context of this problem, these points determine where the cost of materials for the box could be minimized.Solving:\[ 12x - \frac{768}{x^2} = 0 \]we arrive at:\[ x^3 = 64 \quad \Rightarrow \quad x = 4 \text{ feet} \]This calculation indicates a point where the cost function could have a minimum. Once the critical point is found, substitute back to the original equations to ensure the calculated dimensions (here, \( x = 4 \) and \( h = 3 \) feet) provide the intended volume and optimize cost. This confirms the practical value of critical points in decision-making.