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  • Derive the formula for the volume of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.

7. Derive the formula for the volume of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.

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Similar to how we find the surface area of the frustum.

The volume of the frustum is given by-   

=Volume of the bigger cone   -   Volume of the smaller cone

  =\ \frac{1}{3} \pi r^2_1 h_1\ -\ \frac{1}{3} \pi r^2_2 (h_1\ -\ h)

  =\ \frac{\pi}{3}\left ( r_1^2 h_1\ -\ r_2^2 (h_1\ -\ h) \right )                                       

  =\ \frac{\pi}{3}\left ( r_1^2 \left ( \frac{hr_1}{r_1\ -\ r_2} \right )\ -\ r_2^2 \left ( \frac{hr_1}{r_1\ -\ r_2}\ -\ h \right ) \right )

 =\ \frac{\pi}{3}\left ( \frac{hr_1^3}{r_1\ -\ r_2}\ -\ \frac{hr_2^3}{r_1\ -\ r_2} \right )

 =\ \frac{\pi}{3}h\left ( \frac{r_1^3\ -\ r_2^3}{r_1\ -\ r_2} \right )

 =\ \frac{1}{3} \pi h\left ( r_1^2\ +\ r_2^2\ +\ r_1r_2 \right )

Posted by

Devendra Khairwa

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