The formula $s=\sqrt{\frac{S A}{6}}$ gives the length of the side, $s$, of a cube with a surface area, SA. How much longer is the side of a cube with a surface area of 480 square meters than a cube with the surface area of 270 square meters?
A. $\sqrt{5} m$
B. $\sqrt{35} m$
C. $\sqrt{210} m$
D. $7 \sqrt{5} m$
Answer (2)
The side of the cube with a surface area of 480 square meters is 5 meters longer than the side of the cube with a surface area of 270 square meters. The calculated answer is option A: 5 m .
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Calculate the side length of the first cube with surface area 480: s 1 = 6 480 = 4 5 .
Calculate the side length of the second cube with surface area 270: s 2 = 6 270 = 3 5 .
Find the difference in side lengths: s 1 − s 2 = 4 5 − 3 5 = 5 .
The side of the cube with a surface area of 480 square meters is 5 m longer than the side of the cube with a surface area of 270 square meters.
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 of two cubes, one with a surface area of 480 square meters and the other with a surface area of 270 square meters.
Calculate Side Length of First Cube First, let's find the side length of the cube with a surface area of 480 square meters. We substitute S A = 480 into the formula: s 1 = 6 480 = 80 = 16 × 5 = 4 5 So, the side length of the first cube is 4 5 meters.
Calculate Side Length of Second Cube Next, let's find the side length of the cube with a surface area of 270 square meters. We substitute S A = 270 into the formula: s 2 = 6 270 = 45 = 9 × 5 = 3 5 So, the side length of the second cube is 3 5 meters.
Calculate the Difference in Side Lengths Now, we find the difference between the side lengths of the two cubes: s 1 − s 2 = 4 5 − 3 5 = ( 4 − 3 ) 5 = 5 Therefore, the side of the cube with a surface area of 480 square meters is 5 meters longer than the side of the cube with a surface area of 270 square meters.
Final Answer The difference in side lengths is 5 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 cubic box with a specific surface area to minimize material usage, you can use this formula to determine the required side length. Similarly, in construction, if you're building a cubic storage unit and know the total surface area you can cover with available materials, you can calculate the side length to maximize the storage volume within the given surface area constraint.
