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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 at most 5 feet 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]\[
\begin{array}{l}
V \ \textless \ w^3 + 8w^2 + 15w \\
V \geq 25
\end{array}
\][/tex]

A. Carla wrote the system correctly.

B. Carla wrote the system incorrectly; the second equation has the wrong inequality symbol.

C. Carla wrote the system incorrectly; the first equation has the wrong inequality symbol.

D. Carla wrote the system incorrectly; the first equation has the wrong volume expression.

Sagot :

GinnyAnsweridn
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.
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