Cone FrustumDate: 08/31/2001 at 13:47:43 From: Mitch Rosewall Subject: Frustum Cone I am trying to calculate the height of a right frustum cone knowing only r, V, and the angle of the side. These are the only constants I have. Can you help me? Date: 08/31/2001 at 15:43:55 From: Doctor Rob Subject: Re: Frustum Cone Thanks for writing to Ask Dr. Math, Mitch. I drew this picture of a vertical cross-section of the cone and its frustum: V o /|\ / | \ / | \ / x| \ / | \ D o-----o-----o F / r |E \ / | \ / h| \ / | \ /A R | \ o-----------o-----------o A C B You know R, the measure of <A, and the volume V. Now trigonometry and the formula for the volume come to our aid: tan(A) = (x+h)/R = x/r, x = R*tan(A) - h, r = R - h*cot(A). Then V = (R^2+R*r+r^2)*Pi*h/3, V = (R^2+R*[R-h*cot(A)]+[R-h*cot(A)]^2)*Pi*h/3. Now this is a cubic equation in h in which all the coefficients are are known from your given data. It can be put into the form h^3 + h^2*[-3R*tan(A)] + h*[3*R^2*tan^2(A)] + [-3*V*tan^2(A)/Pi] = 0. To solve cubic equations, see the following web page from our Frequently Asked Questions (FAQ): Cubic and Quartic Equations http://mathforum.org/dr.math/faq/faq.cubic.equations.html This equation is particularly simple to solve, and the only real root has the value h = R*tan(A) - (R^3*tan^3[A]-3*V*tan^2[A]/Pi)^(1/3). That formula is the answer to your question. Example: If R = 10 cm, V = 500 cm^3, and m(<A) = 60 degrees, then tan(A) = sqrt(3), and so h = 10*sqrt(3) - (3000*sqrt[3]-4500/Pi)^(1/3), = 1.76537 cm. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ Date: 09/03/2001 at 14:20:05 From: Mitch Rosewall Subject: Calculating "R" for Frustum You have helped me find r, but What I need is to find R and H when r, the volume of the frustum, the and angle of the side are known. /|-----r------\ / | \ / H \ / | \ / | \ /----------R------------\ Please help me. Mitch Date: 09/04/2001 at 10:57:05 From: Doctor Rob Subject: Re: Calculating "R" for Frustum Thanks for writing back, Mitch. The question I answered was how to find H given R, the volume, and the angle. If you know r, V and the angle A, the previous answer contained the following formulas: tan(A) = (x+H)/R = x/r, x = R*tan(A) - H, r = R - H*cot(A), H = (R-r)*tan(A). Then V = (R^2+R*r+r^2)*Pi*H/3, = (R^2+R*r+r^2)*Pi*(R-r)*tan(A)/3, = (R^3-r^3)*Pi*tan(A)/3. Now this is a cubic equation in R in which all the coefficients are are known from your given data. It can be put into the form R^3 = r^3 + 3*V*cot(A)/Pi. The only real root has the value R = (r^3+3*V*cot[A]/Pi)^(1/3). That formula is the answer to your first question. Once you know R, you can find H = (R-r)*tan(A), = -r*tan(A) + (r^3*tan^3[A]+3*V*tan^2[A]/Pi)^(1/3). That formula is the answer to your second question. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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