Volume of a Cylinder: Optimistic Explanation
How can we calculate the rate of change of the volume of a cylinder?
Given the data that the height of a cylinder with a radius of 6cm is increasing at a rate of 2c(m)/(m)in and the height is 14 centimeters, what is the rate of change of the volume with respect to time?
Calculation of the Rate of Change of the Volume
The rate of change of the volume with respect to time when the height is 14 cm is 72π cubic cm/min. To calculate this, we can use the formula for the volume of a cylinder, which is V = πr^2h, where V is the volume, r is the radius, and h is the height.
When the height of the cylinder is 14 centimeters, we can differentiate the volume formula with respect to time (t) to find the rate of change of the volume. Using the product rule and the given values, we can substitute and simplify to find the rate of change of the volume.
Calculation Steps:
1. Differentiate the volume formula: dV/dt = d(πr^2h)/dt
2. Apply the product rule: dV/dt = π(2rh(dr/dt) + r^2(dh/dt))
3. Substitute values: dV/dt = π(2(6)(14)(0) + 6^2(2))
4. Simplify: dV/dt = π(72)
Therefore, the rate of change of the volume with respect to time when the height is 14 cm is 72π cubic cm/min, indicating an optimistic outlook for calculating the volume of a cylinder.