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?

Hint:

When dealing with ratios for geometric figures, always introduce a common multiplier \(k\). Set up the formula for the given property (like volume or area) in terms of \(k\), solve for \(k\), and then substitute it back into the expression for the dimension you need to find.

Answer 1

Step 1: Understanding the Concept:
We are given the volume of a rectangular box and the ratio of its dimensions. We need to express the length in terms of the volume.
Step 2: Detailed Explanation:
Let the length, width, and height of the box be L, W, and H, respectively. The ratio is given as L:W:H = 5:3:2. We can express the dimensions using a common multiplier, \(k\): \[ L = 5k \] \[ W = 3k \] \[ H = 2k \] The volume (V) of the box is given by the formula V = L \(\times\) W \(\times\) H. We are told the volume is "m". \[ m = (5k) \times (3k) \times (2k) \] \[ m = 30k^3 \] Now, we need to solve for the multiplier \(k\) in terms of \(m\): \[ k^3 = \frac{m}{30} \] \[ k = \sqrt[3]{\frac{m}{30}} \] The question asks for the length (L) of the box in terms of \(m\). \[ L = 5k \] Substitute the expression we found for \(k\): \[ L = 5 \sqrt[3]{\frac{m}{30}} \] To get the '5' inside the cube root, we must cube it: \(5 = \sqrt[3]{5^3} = \sqrt[3]{125}\). \[ L = \sqrt[3]{125} \times \sqrt[3]{\frac{m}{30}} = \sqrt[3]{125 \times \frac{m}{30}} \] Simplify the fraction inside the cube root: \[ L = \sqrt[3]{\frac{125m}{30}} = \sqrt[3]{\frac{25m}{6}} \] Step 3: Final Answer:
The length of the box in feet is \( \sqrt[3]{\frac{25m}{6}} \).