Given Information: Given: Surface area is increasing at $9\pi$ square meters per hour, height $h = 3$ meters, surface area at a certain instant is $36\pi$ square meters.
Surface Area Formula: Use the formula for the surface area of a cylinder: Surface area = $2\pi r^2 + 2\pi r h$ .
Calculate Radius: Plug in the given surface area and height to find the radius: $36\pi = 2\pi r^2 + 2\pi r\cdot 3$ .
Find Rate of Change: Simplify the equation: $36\pi = 2\pi r^2 + 6\pi r$ .
Volume Formula: Divide by $2\pi$ to solve for $r$ : $18 = r^2 + 3r$ .
Differentiate Volume: Rearrange the equation: $r^2 + 3r - 18 = 0$ .
Find Rate of Change of Radius: Factor the quadratic equation: $(r + 6)(r - 3) = 0$ .
Calculate Rate of Change: Find the positive value of $r$ : $r = 3$ meters (since radius can't be negative).
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area $\left(\frac{dS}{dt}\right)$ is $9\pi$ , and we need to find the rate of change of the radius $\left(\frac{dr}{dt}\right)$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ . Plug in the values for $dS/dt$ , $r$ , and $h$ : $9\pi = 4\pi 3 \left(\frac{dr}{dt}\right) + 6\pi \left(\frac{dr}{dt}\right)$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ . Plug in the values for $dS/dt$ , $r$ , and $h$ : $9\pi = 4\pi 3 \left(\frac{dr}{dt}\right) + 6\pi \left(\frac{dr}{dt}\right)$ . Simplify to find $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $1$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ . Plug in the values for $dS/dt$ , $r$ , and $h$ : $9\pi = 4\pi 3 \left(\frac{dr}{dt}\right) + 6\pi \left(\frac{dr}{dt}\right)$ . Simplify to find $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $1$ . Combine like terms: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $2$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ . Plug in the values for $\frac{dS}{dt}$ , $r$ , and $h$ : $9\pi = 4\pi 3 \left(\frac{dr}{dt}\right) + 6\pi \left(\frac{dr}{dt}\right)$ . Simplify to find $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $1$ . Combine like terms: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $2$ . Divide by $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $3$ to solve for $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $5$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ . Plug in the values for $dS/dt$ , $r$ , and $h$ : $9\pi = 4\pi 3 \left(\frac{dr}{dt}\right) + 6\pi \left(\frac{dr}{dt}\right)$ . Simplify to find $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $1$ . Combine like terms: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $2$ . Divide by $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $3$ to solve for $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $5$ . Simplify $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $7$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ . Plug in the values for $\frac{dS}{dt}$ , $r$ , and $h$ : $9\pi = 4\pi 3 \left(\frac{dr}{dt}\right) + 6\pi \left(\frac{dr}{dt}\right)$ . Simplify to find $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $1$ . Combine like terms: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $2$ . Divide by $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $3$ to solve for $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $5$ . Simplify $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $7$ . Now plug in the values for $r$ , $\frac{dr}{dt}$ , and $h$ into the rate of change of volume formula: $\frac{dS}{dt}$ $1$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ . Plug in the values for $dS/dt$ , $r$ , and $h$ : $9\pi = 4\pi 3 \left(\frac{dr}{dt}\right) + 6\pi \left(\frac{dr}{dt}\right)$ . Simplify to find $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $1$ . Combine like terms: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $2$ . Divide by $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $3$ to solve for $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $5$ . Simplify $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $7$ . Now plug in the values for $r$ , $\frac{dr}{dt}$ , and $h$ into the rate of change of volume formula: $\frac{dS}{dt}$ $1$ . Simplify the expression: $\frac{dS}{dt}$ $2$ .
Final Rate of Change: Use the formula for the volume of a cylinder: $\text{Volume} = \pi r^2 h$ . Differentiate the volume with respect to time to find the rate of change of volume: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ . We know the rate of change of the surface area ( $\frac{dS}{dt}$ ) is $9\pi$ , and we need to find the rate of change of the radius ( $\frac{dr}{dt}$ ). Differentiate the surface area with respect to time: $\frac{dS}{dt} = 2\pi 2r \left(\frac{dr}{dt}\right) + 2\pi h \left(\frac{dr}{dt}\right)$ . Plug in the values for $dS/dt$ , $r$ , and $h$ : $9\pi = 4\pi 3 \left(\frac{dr}{dt}\right) + 6\pi \left(\frac{dr}{dt}\right)$ . Simplify to find $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $1$ . Combine like terms: $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $2$ . Divide by $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $3$ to solve for $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $5$ . Simplify $\frac{dr}{dt}$ : $\frac{dV}{dt} = \pi 2r \left(\frac{dr}{dt}\right) h$ $7$ . Now plug in the values for $r$ , $\frac{dr}{dt}$ , and $h$ into the rate of change of volume formula: $\frac{dS}{dt}$ $1$ . Simplify the expression: $\frac{dS}{dt}$ $2$ . Calculate the rate of change of volume: $\frac{dS}{dt}$ $3$ cubic meters per hour.