Let's dissect the given problem step-by-step:
1. Understanding the Variables and Relationships:
- The width (w) of the prism is our primary independent variable.
- The height (h) of the prism is 3 feet more than the width, so [tex]\( h = w + 3 \)[/tex].
- The length (l) of the prism is at most 5 feet more than the width, so [tex]\( l = w + 5 \)[/tex].
2. Volume Formula for a Rectangular Prism:
The volume [tex]\( V \)[/tex] of the rectangular prism is given by the product of its length, width, and height:
[tex]\[
V = l \cdot w \cdot h = (w + 5) \cdot w \cdot (w + 3)
\][/tex]
3. Simplifying the Volume Expression:
We now simplify the volume expression [tex]\( V \)[/tex]:
[tex]\[
V = w \cdot (w + 5) \cdot (w + 3)
\][/tex]
Expanding this:
[tex]\[
V = w \cdot (w^2 + 8w + 15) = w^3 + 8w^2 + 15w
\][/tex]
4. Given Inequalities:
According to the problem statement, Carla wrote the following system of inequalities:
[tex]\[
\begin{array}{l}
V < w^3 + 8w^2 + 15w \\
V \geq 25
\end{array}
\][/tex]
5. Checking the Systems:
- First part: [tex]\( V < w^3 + 8w^2 + 15w \)[/tex]
Looking at our derived volume expression, we have:
[tex]\[
V = w^3 + 8w^2 + 15w
\][/tex]
Thus, stating [tex]\( V < w^3 + 8w^2 + 15w \)[/tex] is indeed inaccurate, as [tex]\( V \)[/tex] actually equals [tex]\( w^3 + 8w^2 + 15w \)[/tex], it should rather be [tex]\( V = w^3 + 8w^2 + 15w \)[/tex].
6. Verifying the Volume Constraint:
The minimum volume is given as 25 cubic feet, which translates correctly to:
[tex]\[
V \geq 25
\][/tex]
Confirming this, if we calculate with [tex]\( w = 1 \)[/tex]:
[tex]\[
V = (1 + 5) \cdot 1 \cdot (1 + 3) = 6 \cdot 1 \cdot 4 = 24
\][/tex]
which verifies that for [tex]\( w = 1 \)[/tex], [tex]\( V = 24 \)[/tex] and this indeed does not satisfy [tex]\( V \geq 25 \)[/tex]. To meet the requirement, [tex]\( w \)[/tex] needs to be increased.
7. Conclusion:
Based on the analysis:
- Carla's expression [tex]\( w^3 + 8w^2 + 15w \)[/tex] for the volume [tex]\( V \)[/tex] is correct.
- The inequality [tex]\( V < w^3 + 8w^2 + 15w \)[/tex] is incorrect because it should be [tex]\( V = w^3 + 8w^2 + 15w \)[/tex].
From this evaluation, the correct choice in context of Carla's written inequalities regarding their incorrectness is:
A. Carla wrote the system incorrectly, the first equation has the wrong volume expression.
This correctly acknowledges the error in the inequality which should be equality [tex]\( V = w^3 + 8w^2 + 15w \)[/tex] instead of [tex]\( V < w^3 + 8w^2 + 15w \)[/tex].
