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If the width and height of a rectangular prism are each shrunk to one-seventh of the original size but the length remains the same, what is the formula to find the modified surface area?

A. [tex]SA = \frac{2}{49} lw + \frac{1}{7} lh + \frac{2}{7} wh[/tex]

B. [tex]SA = \frac{1}{7} lw + \frac{1}{49} lh + \frac{2}{7} wh[/tex]

C. [tex]SA = \frac{2}{7} lw + \frac{2}{49} lh + \frac{2}{7} wh[/tex]

D. [tex]SA = \frac{2}{7} lw + \frac{2}{7} lh + \frac{2}{49} wh[/tex]

Please select the best answer from the choices provided.


Sagot :

The problem requires us to find the modified surface area of a rectangular prism when its width and height are each shrunk to one-seventh of their original size, while the length remains unchanged.

Let's start with the original formula for the surface area of a rectangular prism, which is:

[tex]\[ \text{SA} = 2lw + 2lh + 2wh \][/tex]

Here's a step-by-step breakdown to find the modified surface area:

1. Original Dimensions:

- Length ([tex]\( l \)[/tex])
- Width ([tex]\( w \)[/tex])
- Height ([tex]\( h \)[/tex])

2. Modified Dimensions:

- Length ([tex]\( l \)[/tex]) remains the same.
- New width ([tex]\( w' \)[/tex]) = [tex]\( \frac{w}{7} \)[/tex]
- New height ([tex]\( h' \)[/tex]) = [tex]\( \frac{h}{7} \)[/tex]

3. Substitute the modified dimensions into the original surface area formula:

- Surface area formula with new dimensions:

[tex]\[ \text{SA}^{\prime} = 2l \left(\frac{w}{7}\right) + 2l \left(\frac{h}{7}\right) + 2 \left(\frac{w}{7}\right) \left(\frac{h}{7}\right) \][/tex]

4. Simplify each term:

- First term: [tex]\( 2l \left(\frac{w}{7}\right) = \frac{2lw}{7} \)[/tex]
- Second term: [tex]\( 2l \left(\frac{h}{7}\right) = \frac{2lh}{7} \)[/tex]
- Third term: [tex]\( 2 \left(\frac{w}{7}\right) \left(\frac{h}{7}\right) = 2 \left(\frac{wh}{49}\right) = \frac{2wh}{49} \)[/tex]

5. Sum up these simplified terms to get the total modified surface area:

[tex]\[ \text{SA}^{\prime} = \frac{2lw}{7} + \frac{2lh}{7} + \frac{2wh}{49} \][/tex]

So, the formula for the modified surface area is:

[tex]\[ \text{SA}^{\prime} = \frac{2lw}{7} + \frac{2lh}{7} + \frac{2wh}{49} \][/tex]

Comparing with the provided options, the correct answer is:

[tex]\[ \boxed{D. \text{SA} = \frac{2}{7} lw + \frac{2}{7} lh + \frac{2}{49} wh} \][/tex]