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The volume of a rectangular prism is a minimum of 25 cubic feet. The height of the prism is 3 feet more than its width, and its length is more than the width.

Carla wrote this system of inequalities to represent this situation, where [tex]$V$[/tex] is the volume of the prism and [tex]$w$[/tex] is the width.

[tex]
\begin{array}{l}
V \leq w^3 + 8w^2 + 15w \\
V \geq 25
\end{array}
[/tex]

A. Carla wrote the system incorrectly; the first equation has the wrong volume expression.
B. Carla wrote the system correctly.
C. Carla wrote the system incorrectly; the second equation has the wrong inequality symbol.
D. Carla wrote the system incorrectly; the first equation has the wrong inequality symbol.



Answer :

GinnyAnswer
To determine the correctness of Carla's system of inequalities, let's analyze each part of the given situation and the inequalities step-by-step:

### Problem Breakdown

1. Volume of a Rectangular Prism:
- The volume [tex]\(V\)[/tex] of a rectangular prism is given by the product of its width [tex]\(w\)[/tex], height [tex]\(h\)[/tex], and length [tex]\(l\)[/tex].
- Volume formula: [tex]\( V = w \times h \times l \)[/tex].

2. Given Conditions:
- Minimum volume [tex]\( V \geq 25 \)[/tex] cubic feet.
- Height [tex]\( h = w + 3 \)[/tex] (since the height is 3 feet more than the width).
- Length [tex]\( l > w \)[/tex] (length is greater than the width).

### Carla's Inequalities

Carla proposed the following system of inequalities:
[tex]\[ \begin{array}{l} V < w^3 + 8 w^2 + 15 w \\ V \geq 25 \end{array} \][/tex]

### Analyzing Inequalities

1. First Inequality:
[tex]\[ V < w^3 + 8 w^2 + 15 w \][/tex]
- To verify if this inequality correctly represents the volume, let's consider a plausible volume expression.
- We know:
- Height [tex]\( h = w + 3 \)[/tex].
- Assume length as a function of width, e.g., [tex]\( l = w + k \)[/tex] where [tex]\( k > 0 \)[/tex].

- A possible volume expression for the given problem might be [tex]\( V = w \times (w + 3) \times (w + k) \)[/tex].

2. Second Inequality:
[tex]\[ V \geq 25 \][/tex]
- This clearly states the volume should be at least 25 cubic feet, which seems to match the given problem conditions without any ambiguities.

### Correctness of Carla's Inequalities

1. First Inequality Evaluation:
- The expression [tex]\( w^3 + 8w^2 + 15w \)[/tex]:
- Factoring and comparison with a sample volume expression, it appears to be consistent with reasonable volume expectations for a range of [tex]\(w\)[/tex].
- However, given that it matches the plausible functional behavior for a range of [tex]\(w\)[/tex], it seems appropriate.

2. Second Inequality Evaluation:
- [tex]\( V \geq 25 \)[/tex] is directly aligned with the problem statement.

### Conclusion

After reviewing the given conditions and Carla's inequalities:

- A: Incorrect, the volume expression seems aligned.
- B: Correct, the inequalities properly represent the volume constraints and minimum condition.
- C: Incorrect, the second inequality correctly ensures the minimum volume.
- D: Incorrect, the first inequality has the correct symbol comparing volume expressions.

Therefore, the correct statement is:

B. Carla wrote the system correctly.

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