Answered

For all your questions, big or small, IDNLearn.com has the answers you need. Find in-depth and accurate answers to all your questions from our knowledgeable and dedicated community members.

Select the correct answer.

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.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and see you next time for more reliable information.