Question

The interior of a rectangular box is designed by a certain manufacturer to have a volume of \( m \) cubic feet and ratio of length to width to height of 5:3:2. In term of \( m \), which of the following equals the length of the box in feet?

• A. \( \sqrt{\frac{25m}{6}} \)
• B. \( \frac{25}{6} \sqrt{m} \)
• C. \( \sqrt{\frac{9m}{10}} \)
• D. \( \sqrt{\frac{4m}{15}} \)
• E. \( \sqrt{m} \)

Hint:

When solving problems involving volume and ratios, express the variables in terms of a common constant and use the formula for volume.

Answer 1

The Correct Option is A

Step 1: Calculate volume using the ratio of dimensions.
Let the length, width, and height of the box be \( 5k, 3k, \) and \( 2k \) respectively, where \( k \) is a constant.
The volume \( V \) of the box is given by: \[ V = \text{length} \times \text{width} \times \text{height} = 5k \times 3k \times 2k = 30k^3 \] We are told that the volume is \( m \), so: \[ 30k^3 = m \quad \Rightarrow \quad k^3 = \frac{m}{30} \quad \Rightarrow \quad k = \sqrt[3]{\frac{m}{30}} \] Step 2: Calculate the length.
The length is \( 5k \), so: \[ \text{Length} = 5 \times \sqrt[3]{\frac{m}{30}} = \sqrt{\frac{25m}{6}} \] Step 3: Conclusion.
The correct answer is (A).