Question

A cone is modified. The radius is reduced by 1/4 and the height is reduced by 1/5. How does the modified volume relate to the initial volume?,

Answer 1

The modified volume is `5/16` that of the original cone.

Explanation:

Suppose the initial cone has radius `r` and height `h`, then the initial volume is given by:

`V_1 = (pir^2)/h`

If the radius of the modified cone is quadrupled then its new radius is `r/4`, and if the height is reduced by 1/5 the new height is `h/5`, so the volume of the modified cone is given by:

`V_2 = (pi(r/4)^2)/(h/5)`
`\ \ \ \ = ( (pi r^2)/16 )(5/h)`
`\ \ \ \ = 5/16 ( (pi r^2)/h )`
`\ \ \ \ = 5/16 V_1`

Hence, the modified volume is `5/16` that of the original cone.