If the dimensions of a solid proportionally increase by a scale factor of [tex]\(\frac{5}{2}\)[/tex], we need to determine by what factor the surface area increases.
To solve this, we should recall the relationship between the scale factor of dimensions and the scale factor of surface area. The surface area of a solid scales with the square of the linear scale factor. Let's break down the steps:
1. Identify the given scale factor: The linear scale factor given is [tex]\(\frac{5}{2}\)[/tex].
2. Calculate the factor by which the surface area increases: Since surface area is proportional to the square of the linear scale factor, we square [tex]\(\frac{5}{2}\)[/tex].
[tex]\[
\left(\frac{5}{2}\right)^2 = \frac{5}{2} \times \frac{5}{2} = \frac{25}{4}
\][/tex]
Thus, the surface area increases by a factor of [tex]\(\frac{25}{4}\)[/tex].
Therefore, the correct answer is [tex]\(\frac{25}{4}\)[/tex].
