A rectangular box with a square base and no top has a volume of 500 cm³, The dimensions of the box that require the least amount of material are
A. 10 x 10 x 5cm B. 4 x 5 x 5cm C. 50 x 5 x 2cm D. 25 x 10 x 2cm E. 10 x 50 x 1cm
Correct Answer: A
Explanation
To find the dimensions of a rectangular box with a square base and no top that requires the least amount of material, follow these steps:
1. Define the Variables: - Let \( x \) be the length of each side of the square base. - Let \( h \) be the height of the box.
2. Volume Constraint: The volume \( V \) of the box is given by: \[ V = x^2 \cdot h = 500 \text{ cm}^3 \] Thus: \[ h = \frac{500}{x^2} \]
3. Surface Area Calculation: The surface area \( S \) of the box with no top is: \[ S = x^2 + 4 \cdot (x \cdot h) \] Substitute \( h \) from the volume constraint: \[ S = x^2 + 4 \cdot \left(x \cdot \frac{500}{x^2}\right) \] Simplify: \[ S = x^2 + \frac{2000}{x} \]
4. Minimize the Surface Area: To find the value of \( x \) that minimizes \( S \), take the derivative of \( S \) with respect to \( x \) and set it to zero: \[ \frac{dS}{dx} = 2x - \frac{2000}{x^2} \] Set the derivative to zero: \[ 2x - \frac{2000}{x^2} = 0 \] \[ 2x^3 = 2000 \] \[ x^3 = 1000 \] \[ x = \sqrt[3]{1000} = 10 \]
5. Calculate \( h \): Substitute \( x = 10 \) into the volume equation: \[ h = \frac{500}{10^2} = \frac{500}{100} = 5 \]
6. Dimensions: The dimensions of the box that minimize the surface area are: \[ 10 \text{ cm} \times 10 \text{ cm} \text{ base} \times 5 \text{ cm} \text{ height} \]
The dimensions of the box that require the least amount of material are 10 cm x 10 cm x 5 cm.