Respuesta :
Answer:
[tex]157\text{cm}^3/\text{min}[/tex]
Step-by-step explanation:
GIVEN: The height of a cylinder with a fixed radius of [tex]10 \text{cm}[/tex] is increasing at the rate of [tex]0.5\text{cm/min}[/tex].
TO FIND: rate of change of the volume of the cylinder (with respect to time) when the height is [tex]30\text{cm}[/tex].
SOLUTION:
Let the height of cylinder be [tex]=\text{h}[/tex]
Let the volume of cylinder be [tex]=\text{V}[/tex]
radius of cylinder is [tex]=10\text{cm}[/tex]
We know that
Volume of Cylinder [tex]\text{V}=\pi \text{r}^2\text{h}[/tex]
rate of change of height is [tex]\frac{d\text{h}}{dt}=0.5\text{cm/min}[/tex]
rate of change of volume is [tex]\frac{d\text{V}}{dt}[/tex]
rate of change of volume when height is [tex]30\text{cm}[/tex]
[tex]\frac{d\text{V}}{dt}_{\text{h}=30}[/tex][tex]=0.5\pi \text{r}^2[/tex]
putting values
[tex]\frac{d\text{V}}{dt}_{\text{h}=30}[/tex][tex]=0.5\times3.14\times100[/tex]
[tex]\frac{d\text{V}}{dt}_{\text{h}=30}[/tex] [tex]=157\text{cm}^3/\text{min}[/tex]
The rate of change of volume when height is [tex]30\text{cm}[/tex] is [tex]157\text{cm}^3/\text{min}[/tex]
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
