The formula $s=\sqrt{\frac{S A}{6}}$ gives the length of the side, $s$, of a cube with a surface area, $S A$. How much longer is the side of a cube with a surface area of 180 square meters than a cube with the surface area of 120 square meters?
A. $\sqrt{30}-4 \sqrt{5} m$
B. $\sqrt{30}-2 \sqrt{5} m$
C. $\sqrt{10} m$
D. $2 \sqrt{15} m$
Answer (2)
Calculate the side length of the first cube with a surface area of 180 square meters: s 1 = 6 180 = 30 .
Calculate the side length of the second cube with a surface area of 120 square meters: s 2 = 6 120 = 2 5 .
Find the difference between the side lengths: s 1 − s 2 = 30 − 2 5 .
State the final answer: 30 − 2 5 m
Explanation
Problem Analysis We are given the formula s = 6 S A which relates the side length s of a cube to its surface area S A . We need to find the difference in side lengths between two cubes, one with a surface area of 180 square meters and the other with a surface area of 120 square meters.
Calculate Side Length of First Cube First, let's find the side length of the cube with a surface area of 180 square meters. Using the formula, we have: s 1 = 6 180 = 30 So, the side length of the first cube is 30 meters.
Calculate Side Length of Second Cube Next, let's find the side length of the cube with a surface area of 120 square meters. Using the formula, we have: s 2 = 6 120 = 20 = 4 × 5 = 2 5 So, the side length of the second cube is 2 5 meters.
Calculate the Difference in Side Lengths Now, we need to find the difference between the side lengths of the two cubes: s 1 − s 2 = 30 − 2 5 Thus, the difference in side lengths is 30 − 2 5 meters.
Final Answer The side of the cube with a surface area of 180 square meters is 30 − 2 5 meters longer than the side of the cube with a surface area of 120 square meters.
Examples
Understanding the relationship between a cube's surface area and its side length is useful in various real-world scenarios. For example, if you're designing packaging for a product and need a box in the shape of a cube with a specific surface area to minimize material usage, you can use this formula to determine the exact side length required. This ensures you use the least amount of material while still meeting the volume requirements for the product. Similarly, in construction, if you're building a cubic storage container and know the desired surface area, you can calculate the necessary side length to optimize space and material costs.
We calculated the side lengths of two cubes using the formula for side length based on surface area. The difference in side lengths results in 30 − 2 5 meters. Therefore, the chosen answer is B. 30 − 2 5 m .
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