Question

a) A cylindrical length of wire has a radius of 4 mm and a length of 10 cm. If the length is growing at a rate of 2 cm/sec and the radius is shrinking at a rate of 1 mm/sec, what is the rate of change of the volume in cm3/sec at that point in time. (Be careful of units)

b) Consider the same length of wire as before (radius of 4 mm and length of 10 cm). This time we are stretching the wire at a rate of 2 cm/sec. If the total volume is not changing, what is the rate of change of the radius in mm/sec at this point in time.

Answer 1

a)

we know that volume of a cylindrical wire is given by,

where,

r = radius of the wire

l = length of the wire

differentiate both the side with respect to t we can write,

    --------------------------------------------------1)

As given wire has a radius of 4 mm hence,

As given wire has a length of 10 cm hence,

As given length is growing at a rate of 2 cm/sec hence,

As given radius is shrinking at a rate of 1 mm/sec hence,

minus sign indicates radius is shrinking

Hence we have,

Hence put r = 0.4, l = 10, dr/dt = -0.1 and dl/dt = 2 in equation 1) we can write,

As radius and length is in cm and rate of change of radius and rate of change of length is in cm/sec we can write rate of change of volume is in cm³ per second

Hence we can write volume is changing at a rate of,

b)

we know that volume of a cylindrical wire is given by,

where,

r = radius of the wire

l = length of the wire

differentiate both the side with respect to t we can write,

    --------------------------------------------------2)

As given wire has a radius of 4 mm hence,

As given wire has a length of 10 cm hence,

As given we are stretching a wire at a rate of 2 cm/sec hence length is growing at a rate of 2 cm/sec hence,

As given volume is not changing hence rate of change of volume is 0 means dV/dt = 0

Hence we have,

Hence put r = 4, l = 100, dl/dt = 20 and dV/dt = 0 in equation 2) we can write,

As radius and length is in mm and rate of change of length is in mm/sec we can write rate of change of radius is mm/sec

Hence,

minus sign indicates radius is shrinking

Hence we can write radius is shrinking at a rate of 0.4 mm/sec such that volume is constant