CAIE P1 2023 June — Question 9

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2023
SessionJune
TopicConnected Rates of Change

9 Water is poured into a tank at a constant rate of \(500 \mathrm {~cm} ^ { 3 }\) per second. The depth of water in the tank, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\). When the depth of water in the tank is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the tank is given by the formula \(V = \frac { 4 } { 3 } ( 25 + h ) ^ { 3 } - \frac { 62500 } { 3 }\).
  1. Find the rate at which \(h\) is increasing at the instant when \(h = 10 \mathrm {~cm}\).
  2. At another instant, the rate at which \(h\) is increasing is 0.075 cm per second. Find the value of \(V\) at this instant.