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 1,200 square inches than a cube with the surface area of 768 square inches?
A. $\sqrt{2}$ in.
B. $2 \sqrt{2} in$.
C. $4 \sqrt{2} in$.
D. $36 \sqrt{2} in$.
Answer (2)
The side length of the cube with a surface area of 1,200 square inches is 2 2 inches longer than the side of the cube with a surface area of 768 square inches. This difference is calculated using the formula for side length based on surface area. The chosen option is B: 2 2 in.
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Calculate the side length of the first cube with surface area 1200 square inches: s 1 = 6 1200 = 10 2 .
Calculate the side length of the second cube with surface area 768 square inches: s 2 = 6 768 = 8 2 .
Find the difference between the side lengths: s 1 − s 2 = 10 2 − 8 2 = 2 2 .
The side of the cube with a surface area of 1,200 square inches is 2 2 inches longer than the side of the cube with a surface area of 768 square inches: 2 2 in.
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 1200 square inches and another with a surface area of 768 square inches.
Calculate the side length of the first cube First, let's find the side length of the cube with a surface area of 1200 square inches. Using the formula, we have: s 1 = 6 1200 = 200 = 100 × 2 = 10 2 So, the side length of the first cube is 10 2 inches.
Calculate the side length of the second cube Next, let's find the side length of the cube with a surface area of 768 square inches. Using the formula, we have: s 2 = 6 768 = 128 = 64 × 2 = 8 2 So, the side length of the second cube is 8 2 inches.
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 = 10 2 − 8 2 = ( 10 − 8 ) 2 = 2 2 Therefore, the difference in side lengths is 2 2 inches.
Final Answer The side of the cube with a surface area of 1,200 square inches is 2 2 inches longer than the side of the cube with a surface area of 768 square inches.
Examples
Imagine you're designing two differently sized cubic storage containers. Knowing their surface areas, this calculation helps determine the difference in their side lengths, which is crucial for optimizing space and fitting them into specific areas. For instance, if one container has a surface area of 1200 sq. inches and another 768 sq. inches, the difference in their side lengths would be 2 2 inches, guiding your design and placement decisions.
