Question

The height of a cylinder with a fixed radius of 10 cm is increasing at the rate of 0.5 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 30cm. (4 points)

Answer 1

The other answer on here is correct, I just want to keep people from making the same dumb mistake that I did by saying 157= 50pi

Answer 2

`157\text{cm}^3/\text{min}`

Explanation:

GIVEN: The height of a cylinder with a fixed radius of
`10 \text{cm}` is increasing at the rate of
`0.5\text{cm/min}`.

TO FIND: rate of change of the volume of the cylinder (with respect to time) when the height is
`30\text{cm}`.

SOLUTION:

Let the height of cylinder be
`=\text{h}`

Let the volume of cylinder be
`=\text{V}`

radius of cylinder is
`=10\text{cm}`

We know that

Volume of Cylinder
`\text{V}=\pi \text{r}^2\text{h}`

rate of change of height is
`\frac{d\text{h}}{dt}=0.5\text{cm/min}`

rate of change of volume is
`\frac{d\text{V}}{dt}`

rate of change of volume when height is
`30\text{cm}`

`\frac{d\text{V}}{dt}_{\text{h}=30}`
`=0.5\pi \text{r}^2`

putting values

`\frac{d\text{V}}{dt}_{\text{h}=30}`
`=0.5*3.14*100`

`\frac{d\text{V}}{dt}_{\text{h}=30}`
`=157\text{cm}^3/\text{min}`

The rate of change of volume when height is
`30\text{cm}` is
`157\text{cm}^3/\text{min}`