How can you find the volume of a hershey kiss using the "disk method?"?

Question

This is a calculus project...so my teacher expects us to use a function that fits the curve of a hershey kiss and use the disk method. Any ideas?

Answer 1

Do the integral



V = ∫ π (r(h))² dh



where h is the height of the Hersey kiss, and r(h) is the profile function of the Hersey kiss, that is, the radius of the cross section at height h.



One function for r(h) you might want to use to roughly model the profile of the Hersey kiss is Cos(h), so that the integral would be



V = ∫ π (Cos(h))² dh = π( (h/2) + (1/4) Sin(2h) )



where the limits are h = 0 to h = π/2



You can adjust the scaling until it looks more like a realistic Hersey kiss.

Answer 2

Everyone uses different vocabulary for this, but I'm assuming the disk method means is just a 3D volume integration technique where you take a function, find the area of a disk, and then find the area of all the disks added up (integral).



So what you need to come up with is some function that models the outline one slice of a hershey kiss in the first quadrant (so it's really just half a slice). Then you can find the area under that curve and rotate it around the y-axis to get the volume.



So let's say f(x) is your function for the hershey kiss outline. Because you want to rotate around the y-axis, you'll need to find f-1(x). So let's say g(y) is that inverse function. Then π(g(y)^2) is the area of each disc (g(y) gives you the x-distance from the y-axis and so is the radius of each disc). You want to integrate that area from the bottom y coordinate (presumably 0) to how high your hershey goes.



So: integral (π*g(y)^2) dy from yMin to yMax

Answer 3

You need the profile of the Hershery's kiss. Depends on how elaborate you want to get, but see if you can project a large shadow of it onto a piece of paper. All you need is the curve for now. Once you get a decent size shadow, trace it.



Now, go get several copies so you can trasha few while you experiment with it.



Take the curve, overlay a decent trnasparent graph papser grid over it. I think you can get this stuff at Staples, or Office Max, of\r the like.



Now, pick a suitable scale, and approximate as many points as you can on the curve.



For the next part you will have to have some method of taking this set of points and converting them into an equation.



If this isn't some high-brow project and a rough fit is ok, you MAY be able to plunk the data into Excel ahd have it fit the data for you. Otherwise you can search online. There just has to be flash programs that will do this kind of thing if it isn't too terribly exotic.



Once you have the curve, then crank out the 'disc method'....any calc book will show you how that works, and voilà....you're done.

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Answer 6

Hold up the Kiss sideways with the tip pointing to the left. Now doesn't the outline on the upper side look like a nice, yummy chocolatey SINE FUNCTION?



So go with y=r*sin(bx) over the interval 0


The length of the interval pi/2b is the height of the Kiss, so b = pi/2h where h is the height. Your function becomes y=r*sin(pi*x/2h) and the interval is 0


You need to integrate pi*y²*dx (disks of radius y and thickness dx) over this interval.

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