Final answer:
Increasing the volume of a cube by a factor of 7 increases its surface area by 7^(2/3) times, which is less than a factor of 7 because the surface area increases at a slower rate than the volume.
Step-by-step explanation:
The question asks about the effect of increasing the volume of a cube by a factor of 7 on its surface area. We are given that the surface area S of a cube with volume V is S = 6V^(2/3). To find the new surface area after increasing the volume by a factor of 7, we plug in 7V in place of V into the formula. The new surface area S' will then be S' = 6(7V)^(2/3).
When evaluating this, we can simplify as follows:
- S' = 6(7^(2/3) * V^(2/3))
- S' = 6 * 7^(2/3) * V^(2/3)
- S' = 6V^(2/3) * 7^(2/3)
- S' = S * 7^(2/3) since S = 6V^(2/3)
The new surface area is 7^(2/3) times the original surface area, which is an increase, but by less than a factor of 7. This is because the surface area of a cube increases at a slower rate (V^(2/3)) compared to the volume increase (V).