Answer :
To address this problem, let's break down the given situation and the formulations Carla made:
1. Problem Setup:
- Volume of the rectangular prism, [tex]\( V \)[/tex]:
- It has a minimum volume of 25 cubic feet. Hence, [tex]\( V \geq 25 \)[/tex].
- Dimensions:
- The height [tex]\( h \)[/tex] of the prism is 3 feet more than its width [tex]\( w \)[/tex], hence [tex]\( h = w + 3 \)[/tex].
- The length [tex]\( l \)[/tex] of the prism is more than its width [tex]\( w \)[/tex].
2. Volume Expression:
- The volume [tex]\( V \)[/tex] of the rectangular prism can be expressed using the formula:
[tex]\[ V = l \times w \times h \][/tex]
- Given [tex]\( h = w + 3 \)[/tex] and considering the simplest model where [tex]\( l \)[/tex] is also a linear function of [tex]\( w \)[/tex], say [tex]\( l = w + k \)[/tex] for some positive constant [tex]\( k \)[/tex].
3. Carla's Inequalities:
- First Inequality: [tex]\( V < w^3 + 8w^2 + 15w \)[/tex]
- Second Inequality: [tex]\( V \geq 25 \)[/tex]
4. Correcting Carla's System:
- Begin by considering the correct formula based on the dimensions:
[tex]\[ V < w^3 + 8w^2 + 15w \][/tex]
The above inequality encapsulates the requirement that the volume [tex]\( V \)[/tex] generated should be under the cubic expression [tex]\( w^3 + 8w^2 + 15w \)[/tex], derived from a preliminary analysis.
- The minimum volume is specified in the problem:
[tex]\[ V \geq 25 \][/tex]
- Notice that the first inequality given by Carla as [tex]\( V < w^3 + 8w^2 + 15w \)[/tex] should instead represent the correct volume requirement relative to the minimum constraint.
However, we know from the provided true result that the correct form of the inequality which should incorporate the minimum volume [tex]\( V \)[/tex] and the expressions should thus be:
[tex]\[ w^3 + 8w^2 + 15w \geq 25 \][/tex]
Hence, Carla's constructed system of inequalities should ensure equality or upper bound represented in the right form:
5. Final Assessment:
Considering Carla's formulation:
- The first equation [tex]\( V < w^3 + 8w^2 + 15w \)[/tex] has the wrong inequality symbol.
- Therefore, the system is not correctly representing the actual mathematical requirements stated in the problem.
The correct answer is:
D. Carla wrote the system incorrectly: the first equation has the wrong inequality symbol.
1. Problem Setup:
- Volume of the rectangular prism, [tex]\( V \)[/tex]:
- It has a minimum volume of 25 cubic feet. Hence, [tex]\( V \geq 25 \)[/tex].
- Dimensions:
- The height [tex]\( h \)[/tex] of the prism is 3 feet more than its width [tex]\( w \)[/tex], hence [tex]\( h = w + 3 \)[/tex].
- The length [tex]\( l \)[/tex] of the prism is more than its width [tex]\( w \)[/tex].
2. Volume Expression:
- The volume [tex]\( V \)[/tex] of the rectangular prism can be expressed using the formula:
[tex]\[ V = l \times w \times h \][/tex]
- Given [tex]\( h = w + 3 \)[/tex] and considering the simplest model where [tex]\( l \)[/tex] is also a linear function of [tex]\( w \)[/tex], say [tex]\( l = w + k \)[/tex] for some positive constant [tex]\( k \)[/tex].
3. Carla's Inequalities:
- First Inequality: [tex]\( V < w^3 + 8w^2 + 15w \)[/tex]
- Second Inequality: [tex]\( V \geq 25 \)[/tex]
4. Correcting Carla's System:
- Begin by considering the correct formula based on the dimensions:
[tex]\[ V < w^3 + 8w^2 + 15w \][/tex]
The above inequality encapsulates the requirement that the volume [tex]\( V \)[/tex] generated should be under the cubic expression [tex]\( w^3 + 8w^2 + 15w \)[/tex], derived from a preliminary analysis.
- The minimum volume is specified in the problem:
[tex]\[ V \geq 25 \][/tex]
- Notice that the first inequality given by Carla as [tex]\( V < w^3 + 8w^2 + 15w \)[/tex] should instead represent the correct volume requirement relative to the minimum constraint.
However, we know from the provided true result that the correct form of the inequality which should incorporate the minimum volume [tex]\( V \)[/tex] and the expressions should thus be:
[tex]\[ w^3 + 8w^2 + 15w \geq 25 \][/tex]
Hence, Carla's constructed system of inequalities should ensure equality or upper bound represented in the right form:
5. Final Assessment:
Considering Carla's formulation:
- The first equation [tex]\( V < w^3 + 8w^2 + 15w \)[/tex] has the wrong inequality symbol.
- Therefore, the system is not correctly representing the actual mathematical requirements stated in the problem.
The correct answer is:
D. Carla wrote the system incorrectly: the first equation has the wrong inequality symbol.
