Question

I'm not sure if anyone will be able to help me. I've made myself a question. I'm not sure if anyone will be able to help me. I don't even know if it can be solved using a system, or if it is just trial and error.

I want to rotate a graph around the y axis to form a beer glass. I have the equation of the graph, and the shape that it makes MUST hold a total volume of 750mL.

The graph I want to spin around the y axis is:
20ln(x-6)+3

Answer 1

Using the formula:

\[v=\int\limits_{y=0}^{y=a}\pi x^{2}.dx\]

\[750=\int\limits_{y=0}^{y=a}\pi (20\ln(x-6)+3)^{2}.dx\]

I would like the diametre of the glass to be 12 (or there abouts) and hence (x-6) is attached to the ln.

What I am asking is: How tall will the cup be, or how do I solve for a?

Answer 2

I think I've gone about this the wrong way. the graph is
y=20ln(x-6)+3

I need to rearrange the equation for x and then substitute it back into the formula for x, then square it. :\ let me try that

Answer 3

1 thing i noticed is you can't integrate with respect to x if you want to evaluate for y=0 to a
you need to get the function in terms of y first
\[x = e^{(y-3)/20} +6\]

Answer 4

yeah, I managed to pick up my own mistake.

I want to rotate around the y axis. I'm pretty sure I use the formula:

\[750=\int\limits_{0}^{a}\pi(e ^{(y-3)/20)}+6)^2.dy\]

Answer 5

yes, the integrating is straightforward, not sure how difficult isolating "a" will be though
i'll work on it

Answer 6

Would it just be better to use trial and error? :s

Answer 7

Since after general integration, you subtract your initial condition (which is 0) from your final condition (a) you can just do the general integral and then solve for 750=general solution. The problem you are going to have is that you will probably end up with a rather odd answer due to the type of integral you've chosen.

As for real life application. Generally speaking, containers are not filled to the maximum capacity. And, generally you would start with a simple form of the container and then start to use multiple equations to define extraneous parameters

Answer 8

\[750=\int\pi(e ^{(y-3)/20)}+6)^2dy=30 e^{1/30 (-3+y)} π+36 y+C\]

Answer 9

Aaah, that is thinking outside the box.

I also think that as a general application the rotation would be around the x axis and two graphs would be combined in order to form the container (possible log(x) and (x+a)^2 or something).

This was just purely for a fundamental thing, and not for a proposition of real life application.

Answer 10

Yeah, I assumed it was more of a thought experiment, but I figured I'd point out how the application generally occurs :P

Answer 11

in case you are still interested, a is about 4.9

Answer 12

I was, thanks :) appreciated.