Questions: You have been asked to design a rectangular box with a square base and an open top. The volume of the box must be 540 cm^3. The cost of the material for the base is 0.90 per square centimeter and the cost of the material for the sides is 0.10 per square centimeter. Determine the dimensions of the box that will minimize the cost of manufacturing. What is the minimum cost? Give your answer in dollars, rounded to the nearest cent.

You have been asked to design a rectangular box with a square base and an open top. The volume of the box must be 540 cm^3. The cost of the material for the base is 0.90 per square centimeter and the cost of the material for the sides is 0.10 per square centimeter. Determine the dimensions of the box that will minimize the cost of manufacturing. What is the minimum cost? Give your answer in dollars, rounded to the nearest cent.
Transcript text: You have been asked to design a rectangular box with a square base and an open top. The volume of the box must be $540 \mathrm{~cm}^{3}$. The cost of the material for the base is $\$ 0.90$ per square centimeter and the cost of the material for the sides is $\$ 0.10$ per square centimeter. Determine the dimensions of the box that will minimize the cost of manufacturing. What is the minimum cost? Give your answer in dollars, rounded to the nearest cent.
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Solution

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Solution Steps

To solve this problem, we need to minimize the cost of materials for constructing the box while maintaining the given volume. The box has a square base, so we can denote the side length of the base as \( x \) and the height of the box as \( h \). The volume constraint is given by \( x^2 \cdot h = 540 \). The cost function to minimize includes the cost of the base and the sides. The base costs \( 0.90 \times x^2 \) and the sides cost \( 0.10 \times 4xh \). We will express \( h \) in terms of \( x \) using the volume constraint, substitute it into the cost function, and then find the derivative to determine the minimum cost.

Step 1: Define the Variables and Constraints

Let \( x \) be the side length of the square base and \( h \) be the height of the box. The volume constraint is given by:

\[ V = x^2 h = 540 \quad \Rightarrow \quad h = \frac{540}{x^2} \]

Step 2: Formulate the Cost Function

The cost of the materials for the box consists of the cost of the base and the sides. The cost function \( C \) can be expressed as:

\[ C = 0.90 x^2 + 0.10 \cdot 4 x h \]

Substituting \( h \) from the volume constraint:

\[ C = 0.90 x^2 + 0.10 \cdot 4 x \left(\frac{540}{x^2}\right) = 0.90 x^2 + \frac{2160}{x} \]

Step 3: Minimize the Cost Function

To find the minimum cost, we differentiate \( C \) with respect to \( x \) and set the derivative equal to zero:

\[ \frac{dC}{dx} = 1.8 x - \frac{2160}{x^2} = 0 \]

Solving for \( x \) gives:

\[ 1.8 x^3 = 2160 \quad \Rightarrow \quad x^3 = \frac{2160}{1.8} \quad \Rightarrow \quad x \approx 4.9324 \]

Step 4: Calculate the Corresponding Height

Using the value of \( x \) to find \( h \):

\[ h = \frac{540}{(4.9324)^2} \approx 22.1959 \]

Step 5: Calculate the Minimum Cost

Substituting \( x \) back into the cost function:

\[ C \approx 0.90 (4.9324)^2 + \frac{2160}{4.9324} \approx 65.6878 \]

Final Answer

The dimensions of the box that minimize the cost are approximately \( x \approx 4.9324 \) cm and \( h \approx 22.1959 \) cm. The minimum cost is approximately \( 65.69 \) dollars.

Thus, the final answers are:

\[ \boxed{x \approx 4.9324} \quad \text{and} \quad \boxed{h \approx 22.1959} \quad \text{with minimum cost} \quad \boxed{65.69} \]

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