The picture above has all the information about the cone that is given. L is the length between R and 2R (not the side of the cone).
I don't have the overall length, so how can I find the angle in order to get the side of the cone? Is the total height of the cone 2L (if the top wasn't cut up) because the bottom is 2R, and the top is R, so does that means the cone has been cut in half?
I think I know how to set this up with the normal surface area of a cone equation, but how would I set this up as an integral? I understand the basics of integration but I'm still having trouble with it.
Edit: Someone posted an answer but deleted it before I could ask...
Why would you multiply by z/L in the equation: $2 \pi r = 2 \pi \cdot (R + (2R - R) \cdot \frac{z}{L})$. ?
And thanks for the answers so far, I'm still trying to understand them, please bear with me. I think I might have severely overestimated how much I thought I knew about integration...
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$\begingroup$Hint: You already have enough information to find the angle.
Hints:
If you extended the cone so it was complete you would have similar triangles and find that the length of the whole cone as a multiple of $L$ as you have spotted
You can then find the ratio of the curved surface area of the whole cone to the curved surface area cut off, and so calculate the curved surface area remaining. Remember that for similar shapes, the ratio of areas is the square of the linear ratio
Do not forget the surface of the areas of the two ends.
First you have to calculate the height of your cone:
L² = h² + R²
Then you could define a linear function f(x) = R/h*x + R and calculate the lateral surface with the help of
and dont forget the surfaces of the 2 still missing circles
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