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The side measures of a rectangular prism are tripled. What is the relationship between the surface area of the original prism and the surface area of the new prism?

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Answer:

The new prism has 9 times the surface area of the original prism

Explanation:

There are three ways in which you can answer this question.

The hard way:

The surface area of a rectangular prism is given by
A = 2(LW+ LH+ WH)
where L= length, W= width and H = height)

If we were to triple the sides we would get the new side measures as
3L, 3W, 3H

New surface area becomes:
A' = 2 (3L · 3W + 3L · 3H · 3W· 3H)

A' = 2(9LW + 9 LH + 9 WH)

Factoring out 9 from the brackets we get

A' = 2 · 9 (LW+ LH+ WH)

A'/A = 2 · 9 (LW+ LH+ WH) /2(LW+ LH+ WH)

The common term 2(LW+ LH+ WH) cancels out from numerator and denominator leaving 9 as the answer

A smarter and easy way of doing this

A cube is nothing but a rectangular prism with all sides equal. Let a be the length of a side of the cube

A cube has 6 sides. The surface area of each side = a x a = a²

So total surface area A = 6a²

If each side is tripled, each side becomes 3a.
New surface area A' = 6 (3a)² = 6 (9a²)

A'/A = 6 (9a²)/6(a²) = 9

An even easier way

Again we take a cube. But instead of using a variable, let's assign the side of the cube a length of 1 unit

Surface area A = 6 · 1² = 6

After tripling each side becomes 3 units long
New surface area A' = 6 · 3² = 6.9 = 54

A'/A = 54/6 = 9

Choose whichever method you feel comfortable with

User Anant Gupta
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