To determine whether Carla wrote the system of inequalities correctly, let's carefully analyze the given problem and the representations written by Carla.
1. Understanding the problem:
- The volume of the rectangular prism is a minimum of 25 cubic feet. This gives us the inequality:
[tex]\[
V \geq 25
\][/tex]
- The height (h) of the prism is 3 feet more than its width (w):
[tex]\[
h = w + 3
\][/tex]
- The length (l) of the prism is at most 5 feet more than the width (w):
[tex]\[
l \leq w + 5
\][/tex]
Given these relationships, the volume [tex]\( V \)[/tex] of the rectangular prism can be expressed as:
[tex]\[
V = l \times w \times h = (w + 5) \times w \times (w + 3)
\][/tex]
2. Expression for Volume:
To find the volume expression, we need to expand [tex]\( (w + 5) \times w \times (w + 3) \)[/tex]:
[tex]\[
V = w(w + 5)(w + 3)
\][/tex]
Expanding this:
[tex]\[
V = w \times (w^2 + 8w + 15)
\][/tex]
[tex]\[
V = w^3 + 8w^2 + 15w
\][/tex]
Hence, Carla's given volume expression:
[tex]\[
V < w^3 + 8w^2 + 15w
\][/tex]
is indeed correct for the inequality representing the relationship based on the expansion above and taking [tex]\( V \)[/tex] to be maximized.
3. Verification of Carla's system:
- Inequality 1: [tex]\( V < w^3 + 8w^2 + 15w \)[/tex] matches the derived expression [tex]\( w^3 + 8w^2 + 15w \)[/tex].
- Inequality 2: [tex]\( V \geq 25 \)[/tex] correctly expresses the minimum volume condition.
Hence, Carla wrote the system of inequalities correctly based on all steps and conditions provided. Therefore the correct answer is:
A. Carla wrote the system correctly.
