Final answer:
The rate of change of volume V with respect to radius in a cylinder where the height is equal to the radius is dr/dV = 1 / (3πr²).
Step-by-step explanation:
To find the rate of change of volume V with respect to radius, we need to differentiate the equation for the volume of a cylinder with respect to the radius. The volume V of a cylinder is given by the equation V = πr²h, where r is the radius and h is the height. Since the height of the cylinder is equal to its radius (h = r), we can substitute this value into the equation and simplify it to V = πr³.
To find the derivative of V with respect to r, we differentiate the equation V = πr³ using the power rule for differentiation. The power rule states that if we have a variable raised to a power, we can bring down the power and multiply it by the coefficient. In this case, the coefficient is π and the power is 3, so the derivative of V with respect to r is dV/dr = 3πr².
Therefore, the rate of change of volume V with respect to radius is given by the expression dr/dV = 1 / (3πr²).