Answer to Question #275136 in Trigonometry for Ryan

Question

3. Given a frustum of a right circular cone with a slant height of 9ft ., and the radii of bases are 5ft and 7ft. Find:

a. The lateral area

b. Altitude of the frustum

c. The altitude of the entire cone if the cone removed was replaced

Answer 1

Given,

Slant height of right circular cone $(l_1)=9ft$

Base radius $(r_1)= 5ft$

and $(r_2)=7ft$

a) Lateral area of the frustum $=\pi (r_2-r_1)l_1$

$=3.14 (7-5)\times 9 \\ =3.14\times 2\times 9 \\ =59.52ft^2$

b) Altitude of the frustum

$h'=(\sqrt{9^2-4^2})=\sqrt{77}=8.87ft$

$\cos(\theta)=\frac{r_2-r_1}{l_1} \\ =\frac{7-2}{9} \\ =\frac{2}{9}\\ \Rightarrow \theta = \cos^{-1}(\frac{2}{9}) \\ \Rightarrow \theta = 77.16^\circ$

c) Altitude of the cone complete cone

So, $\tan(\theta)=\frac{h}{7}$

$h=\tan(77.16)\times 7 = 30.7ft$