# Thread: Equations: If r is tripled in SA = 4(pi)(r^2)(h), what will happen to value of SA?

1. ## Equations: If r is tripled in SA = 4(pi)(r^2)(h), what will happen to value of SA?

Can someone explain what I did wrong?

a) In the equation $\color{RawSienna}{SA\, =\, 4 \pi r^2 h,}$ predict what will happen to SA if the radius is tripled and the height remains the same.

b) Prove your answer using actual values. In other words, pick test values to plug in and try, according to the stipulations in (a) above. Does your prediction hold true?

c) Create a summary or explanation that describes how to predict what will happen to one variable in an equation based on what happens to another variable. You may use specific examples to aid in your answer.

2. Originally Posted by evking
Can someone explain what I did wrong?

a) In the equation $\color{RawSienna}{SA\, =\, 4 \pi r^2 h,}$ predict what will happen to SA if the radius is tripled and the height remains the same.

My answer: If the radius is tripled and the height remains the same, I predict the surface area will also be tripled.

b) Prove your answer using actual values. In other words, pick test values to plug in and try, according to the stipulations in (a) above. Does your prediction hold true?

My answer: To prove my prediction, I will use 6 and 4 as my test values:

. . .$SA\, =\, 4 \pi (6^2) (4)\, =\, 576\pi$

Now I will triple the radius:

. . .$SA\, =\, 4 \pi\, \color{red}{(6^2 \cdot 3)} (4)\, =\, 1728\pi$. . .<== here

. . .$1728\, =\, 3 \cdot 576$

My prediction does hold true because the surface area was tripled when the radius was tripled.

c) Create a summary or explanation that describes how to predict what will happen to one variable in an equation based on what happens to another variable. You may use specific examples to aid in your answer.

My answer: What happens to one variable in an equation will happen to the other variable in the equation only if the variables have a direct relation.
Instead of tripling r, you tripled r^2.

In other words, you did:

SA = 4 * Pi * [3 * r^2] * h

SA = 4 * Pi * [3*r]^2 * h

3. Let's go to your test case. If the initial radius is 6, what is 3 times the radius? 18, right?

So the initial surface area =

$4 \pi * 6^2 * 4 = 16 * 36 * \pi = 576 \pi.$

So the expanded surface area =

$4 \pi * 18^2 * 4 = 16 * 324 * \pi = 5184 \pi = 9 * 576 \pi = 3^2 * 576 \pi.$

4. Originally Posted by JeffM
Let's go to your test case. If the initial radius is 6, what is 3 times the radius? 18, right?

So the initial surface area =

$4 \pi * 6^2 * 4 = 16 * 36 * \pi = 576 \pi.$

So the expanded surface area =

$4 \pi * 18^2 * 4 = 16 * 324 * \pi = 5184 \pi = 9 * 576 \pi = 3^2 * 576 \pi.$
So if the radius is tripled the surface area does not get tripled?

5. Working symbolically (i.e., unknown radius and height):

SA = 4 * Pi * r^2 * h

SA = 4 * Pi * (3r)^2 * h

= 4 * Pi * 9 * r^2 * h

= 36 * Pi * r^2 *h

The expression:

36 * Pi * r^2 * h

represents a larger value than the expression:

4 * Pi * r^2 * h

How much larger? Is it three times larger?

6. Originally Posted by evking
Originally Posted by JeffM
… initial surface area = $576 \pi$

… expanded surface area = $9 \cdot 576 \pi$
So if the radius is tripled the surface area does not get tripled?
The expanded area is 9 times bigger than the initial area.

"Tripling" means 3 times bigger, so the answer to your question is, "Yes -- if the radius is tripled, the surface area does not get tripled".