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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.
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