+743 A sphere of radius $4$ inches is inscribed in a cone with a base of radius $8$ inches. In inches, what is the height of the cone? Express your answer as a decimal to the nearest tenth.
+129895 
Draw a line from the top of the cone perpendicular to its base.....this is the height of the cone = CF = h
This forms a right triangle with one leg = FB = 8 and the hypotenuse = BC = sqrt (h^2 + 8^2) = sqrt (h^2 +64)
Next draw a line from the center of the sphere perpendicular to the side of the cone
This forms a right triangle similar to the first one with a leg DE = 4 and a hypotenuse= DC = h - 4
By similar triangles
DE / DC = FB / BC
4 / (h - 4) = 8 / sqrt ( h^2 + 64) cross-multiply
4sqrt (h^2 + 64) = 8 (h-4)
sqrt ( h^2 + 64) = 2(h - 4) square both sides
h^2 + 64 = 4 ( h^2 - 8h + 16)
h^2 + 64 = 4h^2 - 32h + 64
3h^2 - 32h = 0
h ( 3h - 32) = 0
Setting the second factor = 0
3h - 32 = 0
3h = 32
h = 32/ 3 in ≈ 10.7 in
+15000 What is the height of the cone?
Hello bingboy!
\(tan\frac{\propto}{2}=\frac{4}{8}=0.5\\ \propto\ =2\ arctan\ 0.5=53.13^\circ\\ {\color{blue}m=}-\ tan\propto\ =\color{blue}-\frac{4}{3}\)
P(8,0)
\(y=m(x-x_P)+y_P\\ y=-\frac{4}{3}(x-8)+0\\ y=-\frac{4}{3}x+\frac{32}{3}\\ \color{blue}y=-\frac{4}{3}x+10.\overline{6}\)
The height of the cone is 10.67 inches.
!
