Answer :

Given: the radius of a cylinder is increasing at the rate 2 cm/sec and its altitude is decreasing at the rate of 3 cm/sec


To find the rate of change of volume when radius is 3 cm and altitude 5cm


Let V be the volume of the cylinder, r be its radius and h be its altitude at any instant of time ‘t’.


We know volume of the cylinder is


V = r2h


Differentiating this with respect to time we get




Now will apply the product rule of differentiation, i.e.,


, so the above equation becomes,




But given of a cylinder is increasing at the rate 2 cm/sec, i.e., and its altitude is decreasing at the rate of 3 cm/sec, i.e., , by subsitituting the above values in equation (i) we get



When radius of the cylinder, r = 3cm and its altitude, h = 5cm, the equation (ii) becomes,





Hence the rate of change of volume when radius is 3 cm and altitude 5cm is 33 cm3/sec


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