Answered

Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

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

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

Sagot :

GinnyAnswer
To determine the correctness of Carla's system of inequalities, let's break down the given information and form the appropriate expressions step-by-step.

1. Volume of a Rectangular Prism:
The volume [tex]\( V \)[/tex] of a rectangular prism is given by:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
Let's denote:
- the width by [tex]\( w \)[/tex],
- the height by [tex]\( h \)[/tex],
- the length by [tex]\( l \)[/tex].

2. Relationships Provided:
- The height of the prism is 3 feet more than its width.
[tex]\[ h = w + 3 \][/tex]
- The length is at most 5 feet more than the width.
[tex]\[ l \leq w + 5 \][/tex]

3. Volume Expression:
Substituting the height and length into the volume formula:
[tex]\[ V = l \times w \times h \][/tex]
Using the relationships:
[tex]\[ V = (w + 5) \times w \times (w + 3) \][/tex]
Simplify the expression:
[tex]\[ V = (w + 5) \times w \times (w + 3) \][/tex]
Perform the multiplication:
[tex]\[ V = (w^2 + 8w + 15) \times w \][/tex]
Therefore:
[tex]\[ V = w^3 + 8w^2 + 15w \][/tex]

4. Volume Inequality:
Given that the volume is a minimum of 25 cubic feet:
[tex]\[ V \geq 25 \][/tex]

5. Carla’s System Analysis:
Now, let’s examine Carla’s system:
[tex]\[ \begin{array}{l} V < w^3 + 8w^2 + 15w \\ V \geq 25 \end{array} \][/tex]

- The first inequality [tex]\( V < w^3 + 8w^2 + 15w \)[/tex] states that the volume [tex]\( V \)[/tex] should be less than the expression for the volume [tex]\( w^3 + 8w^2 + 15w \)[/tex]. This contradicts the condition as [tex]\( V \)[/tex] should actually be equal to [tex]\( w^3 + 8w^2 + 15w \)[/tex] when the height, width, and length of the prism are considered exactly as given. Therefore, this inequality is incorrect.
- The second inequality [tex]\( V \geq 25 \)[/tex] correctly states that the volume needs to be at least 25 cubic feet.

Based on this analysis:

- The first equation is incorrect because it has the wrong inequality symbol. It should have been an equality: [tex]\( V = w^3 + 8w^2 + 15w \)[/tex].

Thus, the correct answer is:
C. Carla wrote the system incorrectly; the first equation has the wrong inequality symbol.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.