Question

The length,breadth and height of a cuboid depends on time t as
L=1+sint
B=t square -1
H=(t+1)
find the rate of change of volume with time at t=π\2second

Answer 1

volume of cuboid = length × breadth × height

i.e., V = L × B × H

or, V = (1 + sint) × (t² - 1) × (t + 1)

differentiating with respect to time,

dV/dt = cost(t² - 1)(t + 1) + (1 + sint)(2t)(t + 1) + (1 + sint)(t² - 1)(1)

now, putting t = π/2 sec

so, dV/dt = cosπ/2(π²/4 - 1)(π/2 + 1) + (1 + sinπ/2)(2 × π/2)(π/2 - 1) + (1 + sinπ/2)(π²/4 - 1)(1)

= 0 + 2 × π × (π - 2)/2 + 2 × (π² - 4)/4

= π(π - 2) + (π² - 4)/2

= (3π² - 4π - 4)/2 m³/s

hence, rate of change of volume of cuboid is (3π² - 4π - 4)/2 m³/s