To determine how much longer the side of a cube with a surface area of 1,200 square inches is compared to a cube with a surface area of 768 square inches, we will follow these steps:
1. Calculate the side length of the cube with a surface area of 1,200 square inches:
We start by applying the formula [tex]\( s = \sqrt{\frac{SA}{6}} \)[/tex].
For [tex]\( SA = 1200 \)[/tex] square inches:
[tex]\[
s_1 = \sqrt{\frac{1200}{6}} = \sqrt{200} \approx 14.142135623730951
\][/tex]
2. Calculate the side length of the cube with a surface area of 768 square inches:
Again, applying the formula [tex]\( s = \sqrt{\frac{SA}{6}} \)[/tex].
For [tex]\( SA = 768 \)[/tex] square inches:
[tex]\[
s_2 = \sqrt{\frac{768}{6}} = \sqrt{128} \approx 11.313708498984761
\][/tex]
3. Find the difference in side lengths:
Subtract the side length of the cube with [tex]\( SA = 768 \)[/tex] square inches from the side length of the cube with [tex]\( SA = 1200 \)[/tex] square inches.
[tex]\[
\Delta s = s_1 - s_2 = 14.142135623730951 - 11.313708498984761 \approx 2.82842712474619
\][/tex]
4. Interpret the difference in terms of the given options:
The approximate value [tex]\( 2.82842712474619 \)[/tex] corresponds to [tex]\( 2\sqrt{2} \)[/tex]:
[tex]\[
2\sqrt{2} \approx 2 \times 1.414213562 = 2.828427124
\][/tex]
Therefore, the side of the cube with a surface area of 1,200 square inches is [tex]\( 2\sqrt{2} \)[/tex] inches longer than the side of the cube with a surface area of 768 square inches.
The correct answer is:
[tex]\[ \boxed{2 \sqrt{2} \text{ in.}} \][/tex]
