# Thread: Volume of Wine Glass

1. ## Volume of Wine Glass

When a shape lying entirely on one side of the X axis is rotated about that axis it generates a solid of revolution. This investigation explores how solids of revolution and integration can be used to determine the volume of a wine glass.

I have attached photos of what I am struggling. It would be amazing if someone could help me out, showing full working and explanations!

Thanks!

2. ## Re: Volume of Wine Glass

We can help you with parts 1 and 2 if you tell us where you're having difficulty. Assuming you've done parts 1 and 2 with no difficulty, in part 3 you can probably fit a quadratic curve (y = ax^2 + bx + c) to the given shape. You'll need to reserve one parameter for the size - you'll calculate that one after you integrate (so the volume is correct).

- Hollywood

3. ## Re: Volume of Wine Glass

For part 3, i've found a function for the curve. its -0.003578x^4+0.0962x^3-0.9892x^2+0.333x+0.2634
How do i find the volume of the wine glass that it creates?
Thanks

4. ## Re: Volume of Wine Glass

Interesting choice of function. To find the amount of wine it will hold, you integrate $\pi{y}^2$ - so $V=\pi\int_a^b (-0.003578x^4+0.0962x^3-0.9892x^2+0.333x+0.2634)^2 \,dx$, where x=a is the bottom of the bowl of the wine glass and x=b is the rim of the wine glass.

To find the volume of the wine glass, you're finding the surface area and multiplying by 2mm. I'm not sure what you're supposed to do about the stem and the bottom. To get the surface area, you integrate $2\pi{y}\sqrt{1+\left(\frac{dy}{dx}\right)^2}$ - the first part is just the circumference of a circle with radius y, and the square root is there because a more slanted disk has more surface area. You'll have to do this integral numerically.

- Hollywood

5. ## Re: Volume of Wine Glass

I've changed the function to -0.000962x^4+0.0277x^3-0.337x^2+1.85x+0.0685
with that, how do i do part 3 question 3?

Thanks

7. ## Re: Volume of Wine Glass

Thanks
Part 2 says to find the volume of a right circular cone with radius 3 and height 6 by using the cone volume formula and then using a definite integral ...

I assume you can determine the volume using the standard formula.

To set up the integral requires the equation of the line $y=mx$ to be rotated about the x-axis ... that line is the slant-height of the cone and has equation $y=\dfrac{r}{h} \cdot x$.

$\displaystyle V= \pi \int_0^h \left(\dfrac{r}{h} \cdot x \right)^2 \, dx$

The problem provides values for $r$ and $h$ ... can you take it from here?

8. ## Re: Volume of Wine Glass

Hello,
I have the same investigation and I was wondering if anyone could help me with Part 3, question 3 finding the volume of the glass itself if it has to be 2mm thick.

9. ## Re: Volume of Wine Glass

I would use an integral to find the surface area of the bowl, then multiply that result by the desired thickness.

10. ## Re: Volume of Wine Glass

I am a bit confised on how to find the surface area of the wine glass. I don't inderstand the equation in post 4 which shows how to find the surface area. Could someone please explain what each term is!
Thankyou...

11. ## Re: Volume of Wine Glass

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### wine glass equation

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