Question

The radius of a right circular cone is increasing at a rate of 1.6 in/s while its height is decreasing at a rate of 2.4 in/s. At what rate is the volume of the cone changing when the radius is 129 in. and the height is 128 in.?

Answer 1

Step-by-step explanation:

`(dr)/(dt)` = 1.6in/s,
`(dh)/(dt)` = -2.4 in/s radius = 129 in and height = 128 in

the volume of a right circular cone =
`(1)/(3) \pi r^(2) h`

using chain rule equation to determine the rate of change in volume

`(dv)/(dt)` =
`(dv)/(dr)` (
`(dr)/(dt)`) +
`(dv)/(dh)`(
`(dh)/(dt)`)

partial differentiating with respect to radius and height respectively

`(dv)/(dr)` =
`(d)/(dr)`(
`(1)/(3)\pi r^(2) h`) =
`(2)/(3)\pi rh` = 11008 π

`(dv)/(dh)` =
`(d)/(dh)`(
`(1)/(3)\pi r^(2) h`) =
`(1)/(3)\pi r^(2)` = 5547π

`(dv)/(dt)` = 11008 π(1.6 in/s) + 5547π (-2.4in/s) = 4300π in³ / s