Certainly! Let's analyze the problem step-by-step.
Given:
Cary's equation for the surface area of the box:
[tex]\[ 148 = 2(6w + 6h + hw) \][/tex]
After solving for [tex]\( w \)[/tex], Cary obtained:
[tex]\[ w = \frac{74 - 6h}{h + 6} \][/tex]
We need to find which of the given options is equivalent to Cary's expression for [tex]\( w \)[/tex].
Step 1: List the given options for [tex]\( w \)[/tex]:
1. [tex]\( w = \frac{148 - 6h}{12 + h} \)[/tex]
2. [tex]\( w = \frac{148 - 12h}{12 + 2h} \)[/tex]
3. [tex]\( w = 136 - 14h \)[/tex]
4. [tex]\( w = 136 - 10h \)[/tex]
Step 2: Check each option to determine if it is equivalent to [tex]\( w = \frac{74 - 6h}{h + 6} \)[/tex].
Given the result, we can infer:
1. First option: [tex]\( w = \frac{148 - 6h}{12 + h} \)[/tex]
- This option is not equivalent.
2. Second option: [tex]\( w = \frac{148 - 12h}{12 + 2h} \)[/tex]
- This option is equivalent.
3. Third option: [tex]\( w = 136 - 14h \)[/tex]
- This option is not equivalent.
4. Fourth option: [tex]\( w = 136 - 10h \)[/tex]
- This option is not equivalent.
Conclusion:
The equation equivalent to Cary's expression for [tex]\( w \)[/tex] is:
[tex]\[ w = \frac{148 - 12h}{12 + 2h} \][/tex]
Therefore, the correct choice is:
[tex]\[ w = \frac{148 - 12h}{12 + 2h} \][/tex]
