At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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.
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 appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
