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Cary calculated the surface area of a box in the shape of a rectangular prism. She wrote the equation [tex]148 = 2(6w + 6h + hw)[/tex] to represent the width and height of the box. She solved for [tex]w[/tex] and got [tex]w = \frac{74 - 6h}{h + 6}[/tex]. Which of the following is an equivalent equation?

A. [tex]w = \frac{148 - 6h}{12 + h}[/tex]

B. [tex]w = \frac{148 - 12h}{12 + 2h}[/tex]

C. [tex]w = 136 - 14h[/tex]

D. [tex]w = 136 - 10h[/tex]

Sagot :

GinnyAnsweridn
To determine which of the given equations is equivalent to the equation [tex]\( w = \frac{74 - 6h}{h + 6} \)[/tex], we'll need to manipulate each option algebraically to see if we can form or match it to the original equation.

Let's analyze each option step-by-step:

### Option 1: [tex]\( w = \frac{148 - 6h}{12 + h} \)[/tex]

1. We'll start by attempting to manipulate the original equation [tex]\( w = \frac{74 - 6h}{h + 6} \)[/tex] to see if it can be transformed into this form.
2. Multiply both the numerator and the denominator by 2 for the original equation:
[tex]\[ w = \frac{74 - 6h}{h + 6} \cdot \frac{2}{2} = \frac{2(74 - 6h)}{2(h + 6)} = \frac{148 - 12h}{2h + 12}. \][/tex]
3. Simplify the denominator:
[tex]\[ \frac{148 - 12h}{12 + 2h}. \][/tex]
Clearly, this does not match [tex]\( \frac{148 - 6h}{12 + h} \)[/tex].

Thus, Option 1 is not equivalent.

### Option 2: [tex]\( w = \frac{148 - 12h}{12 + 2h} \)[/tex]

1. We already have the manipulation done in Option 1:
[tex]\[ w = \frac{74 - 6h}{h + 6} \cdot \frac{2}{2} = \frac{2(74 - 6h)}{2(h + 6)} = \frac{148 - 12h}{12 + 2h}. \][/tex]

This is an exact match to Option 2, so it is equivalent.

### Option 3: [tex]\( w = 136 - 14h \)[/tex]

1. This is a linear equation and is not in the form of a fraction (rational form), so it doesn't match the structure of the original equation [tex]\( w = \frac{74 - 6h}{h + 6} \)[/tex].

Thus, Option 3 is not equivalent.

### Option 4: [tex]\( w = 136 - 10h \)[/tex]

1. Similar to Option 3, this is also a linear form and is not in the form of a fraction, hence it doesn't match the structure of [tex]\( w = \frac{74 - 6h}{h + 6} \)[/tex].

Thus, Option 4 is not equivalent.

### Conclusion

The only option that matches the equation [tex]\( w = \frac{74 - 6h}{h + 6} \)[/tex] is Option 2. After multiplying both the numerator and the denominator by 2, it matches the form [tex]\( \frac{148 - 12h}{12 + 2h} \)[/tex].

Therefore, the correct equivalent equation is:
[tex]\[ w = \frac{148 - 12h}{12 + 2h}. \][/tex]

So the answer is:
[tex]\[ \boxed{w = \frac{148 - 12h}{12 + 2h}} \][/tex]
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