A right circular cone is cut in half .The diameter of the base is 2r and the perpendicular height is h, By considering the volume of one slice of the half cone, and then summing all such slices, show from first principles that the volume is 1/6pi r^2h
1
Expert's answer
2011-12-09T08:48:50-0500
Consider a slice of the cone, a distance w down from the vertex, of thickness δw
Looking at the cut-face of the cone, you can see that the radius of the slice (s) can be obtained using the idea of similar triangles: <img style="width: 155px; height: 39px;" src="data:image/png;base64,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" alt=""> The cross sectional area of the (semi circular) slice= <img style="width: 45px; height: 43px;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEYAAABDCAIAAACugeRgAAABjklEQVRoge2Zy5XDIAxF0xPd0Ayt0AqlUAmzYGzzNY6So+jZujvnsNCNrBcCr3Q7Xr8u4PuoEgKqhIAqIaBKCKgSAnBKwZkSF7oVYErBGWN9zA/R25EUmFJN9LYw3EBWit6O3jxYpYlPQlWa+yRmpSatOtx8aTExVUQM4FNa+RhjfV5ZT/3gqehPcF23mJSOSuqagutjuMuxo+wc29P+ZbhnaWDU1vRf92xUlnArXfmtLJpB8WJWumR0LCVp8SpVr102OsuuK0t6+EN8L3A0SNHb6pPxluccRqX2Kz+Ugtt61yTgMD5WsCltI19Mxp4C+2d9SBPyAXNDdIoqIfAUpfUOc7nR+iFPUcKGSendrtNQJRyeEg83VMJGleQT3N2Uord3U0ryXrwmbClJKkkp6zRnYtIPvc7pzu8JZ17BGUlKHaRjPFFd6iCdeUlWohxLpiRYaXRNcw2ZSnSfJFLp9CL2AtKUPvVJ0pSit20gBCf45mLJ6CaWsH0QpDT9L42r9C1UCQFVQkCVEFAlBP4AdTkLBbnrh8MAAAAASUVORK5CYII=" alt=""> so the volume may be approximated to that of a half cylinder: <img style="width: 52px; height: 43px;" src="data:image/png;base64,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" alt=""> provided that δw is sufficiently small.
Substituting for s, the volume of the slice (v) is just <img style="width: 127px; height: 48px;" src="data:image/png;base64,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" alt=""> The volume (V) of the whole half cone is just the sum of the volumes of all the slices between w = 0 and w = h <img style="width: 152px; height: 57px;" src="data:image/png;base64,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" alt=""> In the limiting case { as δw tends to 0 } , and taking out the constants π,r²,h²,2 this boils down to <img style="width: 564px; height: 55px;" src="data:image/png;base64,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" alt="">
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