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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]



Answer :

GinnyAnswer
Let's solve this step-by-step:

We start with the given equation:

[tex]\[ 148 = 2(6w + 6h + hw) \][/tex]

First, distribute the 2 on the right-hand side:

[tex]\[ 148 = 12w + 12h + 2hw \][/tex]

We want to solve for [tex]\( w \)[/tex] in terms of [tex]\( h \)[/tex]. To do this, let's isolate [tex]\( w \)[/tex]:

1. Combine like terms on the right-hand side:
[tex]\[ 148 = 12w + 12h + 2hw \][/tex]

2. Move the [tex]\( 12h \)[/tex] term to the left-hand side:
[tex]\[ 148 - 12h = 12w + 2hw \][/tex]

3. Factor [tex]\( w \)[/tex] out of the terms on the right-hand side:
[tex]\[ 148 - 12h = w(12 + 2h) \][/tex]

4. Divide both sides by [tex]\( (12 + 2h) \)[/tex] to isolate [tex]\( w \)[/tex]:
[tex]\[ w = \frac{148 - 12h}{12 + 2h} \][/tex]

Let's compare this to the given equivalent equations:

- [tex]\( w = \frac{148 - 6h}{12 + h} \)[/tex]
- [tex]\( w = \frac{148 - 12h}{12 + 2h} \)[/tex]
- [tex]\( w = 136 - 14h \)[/tex]
- [tex]\( w = 136 - 10h \)[/tex]

Comparing directly, we see that:

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

is exactly equivalent to our derived equation because both the numerator and denominator match.

Thus, the equivalent equation is:

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

So, the correct equation is:

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

Therefore, the correct answer is the second option:

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

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