A right circular cone of height h and base radius r has total surface area S con

Question

A right circular cone of height h and base radius r has total surface area S consisting of its base area plus its side area, leading to the formula: Suppose you start out with a cone of height 8 cm and base radius 6 cm, and you want to change the dimensions in such a way that the total surface area remains the same. Suppose you increase the height by 40/100. In this problem, use tangent line approximation to estimate the new value of r so that the new cone has the same total surface area. The estimated value of r =

Explanation / Answer

s=pi r^2 + pi r sqrt(r^2 +h^2)

h=8, r=6 ==>s=pi 6^2 + pi *6* sqrt(6^2 +8^2)

s=36pi + pi *6*10

s=96pi

96pi=pi r^2 + pi r sqrt(r^2 +h^2)

96 = r^2 + r sqrt(r^2 +h^2)

differentiate with respect to h

0=2r dr/dh + (dr/dh)sqrt(r^2 +h^2) +    [r/(2sqrt(r^2+h^2))]*[2rdr/dh +2h]   ,r=6,h=8

==>2*6 dr/dh + (dr/dh)sqrt(6^2 +8^2) +    [6/(2sqrt(6^2+8^2))]*[2*6dr/dh +2*8]

==>12dr/dh+10dr/dh +(6/20)[12dr/dh +16] =0

==>12dr/dh+10dr/dh +(18/5)dr/dh + (24/5) =0

==>(128/5)dr/dh=-24/5

==>dr/dh=-24/128

==>dr/dh=-3/16

==>dr=(-3/16)dh

==>dr=(-3/16)(40/100)

==>dr=-0.075

new r =r+dr= 6-0.075=5.925

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