# Thread: [SOLVED] Rational Function: Application word problem

1. ## [SOLVED] Rational Function: Application word problem

I need help understanding the process to combine terms to result in a specific function. I have a cylinder with the volume and surface area defined below:

V=Pi(r^2)h
S=2(Pi(r^2)+2(Pi)rh

The book is asking: Assume that the volume of the cylinder is 1000 cm^3. Express the surface area as a function of the radius. After combining terms in your answer, show that the resulting function can be written:

S(r) = (2(Pi)(r^2) + 2000)/r where (r>0)

I understand that the problem is looking for S(r), where 2(Pi)(r^2)+2(Pi)rh is the formula for the surface area, but I don't understand how the book ends up with the resulting function.

Thank you!

2. Hello, Snowcrash!

If you typed everything correctly, their answer is wrong . . .

I have a cylinder with its volume and surface area: .$\displaystyle \begin{array}{cccc}(1) & V &=&\pi r^2h \\ (2) &S &=&2\pi r^2 + 2\pi rh \end{array}$

Assume that the volume of the cylinder is 1000 cm³.
Express the surface area as a function of the radius.
Show that the function can be written: .$\displaystyle S(r) \:= \:\frac{2\pi r^2 + 2000}{r}$ .
. . . This is wrong!
From (1), we have: .$\displaystyle \pi r^2h \:=\:1000 \quad\Rightarrow\quad h \:=\:\frac{1000}{\pi r^2}$

Substitute into (2): .$\displaystyle S \;=\;2\pi r^2 + 2\pi r\left(\frac{1000}{\pi r^2}\right)$

Therefore: .$\displaystyle S \;=\;2\pi r^2 + \frac{2000}{r}$

3. Originally Posted by Snowcrash
I need help understanding the process to combine terms to result in a specific function. I have a cylinder with the volume and surface area defined below:

V=Pi(r^2)h
S=2(Pi(r^2)+2(Pi)rh

The book is asking: Assume that the volume of the cylinder is 1000 cm^3. Express the surface area as a function of the radius. After combining terms in your answer, show that the resulting function can be written:

S(r) = (2(Pi)(r^2) + 2000)/r where (r>0)

I understand that the problem is looking for S(r), where 2(Pi)(r^2)+2(Pi)rh is the formula for the surface area, but I don't understand how the book ends up with the resulting function.

Thank you!
You have V=1000
and $\displaystyle V=\pi r^2h$

hence, $\displaystyle 1000=\pi r^2 h$

Can you continue now?

EDIT: beaten by Soroban....he will be glad.

4. Thank you Soroban and Malay -

I double-checked and the function that I did type the function incorrectly, in that I used a square instead of cube (2(Pi)(r^3):

S(r) = (2(Pi)(r^3) + 2000)/r where (r>0)

Would the cube come in because the volume is 1000cm^3??

My first instinct would be to solve the volume formula for the radius because the problem is asking me to "express the surface area as a function of the radius" or s(r) instead of solving for the h value.

Like r^2 = 1000/(Pi(h))

But you are solving for h instead - how did you know to solve for h?

5. Originally Posted by Snowcrash
Thank you Soroban and Malay -

I double-checked and the function that I did type the function incorrectly, in that I used a square instead of cube (2(Pi)(r^3):

S(r) = (2(Pi)(r^3) + 2000)/r where (r>0)

Would the cube come in because the volume is 1000cm^3??

My first instinct would be to solve the volume formula for the radius because the problem is asking me to "express the surface area as a function of the radius" or s(r) instead of solving for the h value.

Like r^2 = 1000/(Pi(h))

But you are solving for h instead - how did you know to solve for h?
Hi Snowcrash,

You said you needed the Surface Area stated as a function of the radius.

Solving for h in the Volume formula and substituting this into the Surface Area formula gave you the Surface Area as a function of the radius. You are not solving for r. You're just stating the function in terms of r.

$\displaystyle S(r)=2\pi r^2+\frac{2000}{r}$

$\displaystyle \pi$ is a constant.

6. Okay, that does make sense. However, I'm still unsure of why the book shows the final formula with a cube:

$\displaystyle S(r) \:= \:\frac{2\pi r^3 + 2000}{r}$

Soroban said that the answer is incorrect - but I wrote it incorrectly with $\displaystyle S(r) \:= \:\frac{2\pi r^2 + 2000}{r}$

How does the cube get in there? Is it really incorrect or is it because the 1000CM is 1000CM^3?

7. Originally Posted by Snowcrash
Okay, that does make sense. However, I'm still unsure of why the book shows the final formula with a cube:

$\displaystyle S(r) \:= \:\frac{2\pi r^3 + 2000}{r}$

Soroban said that the answer is incorrect - but I wrote it incorrectly with $\displaystyle S(r) \:= \:\frac{2\pi r^2 + 2000}{r}$

How does the cube get in there? Is it really incorrect or is it because the 1000CM is 1000CM^3?
$\displaystyle \frac{2\pi r^3 + 2000}{r}$ $\displaystyle =\frac{2\pi r^3}{r}+\frac{2000}{r}=2\pi r^2+\frac{2000}{r}$

8. Originally Posted by Snowcrash
Okay, that does make sense. However, I'm still unsure of why the book shows the final formula with a cube:

$\displaystyle S(r) \:= \:\frac{2\pi r^3 + 2000}{r}$

Soroban said that the answer is incorrect - but I wrote it incorrectly with $\displaystyle S(r) \:= \:\frac{2\pi r^2 + 2000}{r}$

How does the cube get in there? Is it really incorrect or is it because the 1000CM is 1000CM^3?
Snowcrash,

$\displaystyle S=2\pi r^2+\frac{2000}{r}$ can be rewritten as

$\displaystyle S=\frac{2\pi r^3+2000}{r}$

It's just another way of expressing the same function. All we did here was put the right hand expression over one denominator.
$\displaystyle S=2\pi r^2+\frac{2000}{r} \implies S=\frac{2\pi r^3}{r}+\frac{2000}{r} \implies S=\frac{2\pi r^3+2000}{r}$

9. Thanks, I see it now. Thank you for your help everyone!