Forming differential equations and connected rates of changeEdexcel International A Level Maths: Subtopic test

1. 1
   At time $t$ minutes a leaking tank holds $V$ m$^3$ of water. Water leaks out at a rate proportional to the volume $V$ of water remaining in the tank.

   (a)

   Which differential equation models the volume $V$, where $k$ is a positive constant?

   [1 mark]

   • A$\frac{dV}{dt}=kV$
   • B$\frac{dV}{dt}=-\frac{k}{V}$
   • C$\frac{dV}{dt}=-kt$
   • D$\frac{dV}{dt}=-kV$

   (b)

   When $V=200$ the volume is decreasing at $5$ m$^3$ per minute. Find $k$.

   [1 mark]

   • A$40$
   • B$0.025$
   • C$-0.025$
   • D$1000$

   (c)

   Find the rate at which the volume is changing when $V=80$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A spherical balloon is inflated so that its radius $r$ cm increases at a constant rate of $0.5$ cm s$^{-1}$. The volume of a sphere is $V=\frac43\pi r^3$ and its surface area is $S=4\pi r^2$.

   (a)

   Find $\frac{dV}{dr}$.

   [1 mark]

   • A$4\pi r^2$
   • B$\frac43\pi r^2$
   • C$\frac{1}{3}\pi r^4$
   • D$4\pi r$

   (b)

   Find the rate at which the volume is increasing when $r=6$.

   [1 mark]

   • A$144\pi$ cm$^3$ s$^{-1}$
   • B$288\pi$ cm$^3$ s$^{-1}$
   • C$72\pi$ cm$^3$ s$^{-1}$
   • D$12\pi$ cm$^3$ s$^{-1}$

   (c)

   Find the rate at which the surface area is increasing when $r=6$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Liquid is poured at a constant rate of $20$ cm$^3$ s$^{-1}$ into an inverted cone with its axis vertical. At time $t$ seconds the depth of liquid is $h$ cm and its volume is $V=\frac13\pi h^3$ cm$^3$.

   (a)

   Find the rate at which the depth is increasing when $h=5$.

   [3 marks]

   (b)

   The liquid now also leaks out through a hole at a rate of $kh$ cm$^3$ s$^{-1}$, where $k$ is a constant, while liquid is still poured in at $20$ cm$^3$ s$^{-1}$. (i) Show that $\frac{dh}{dt}=\frac{20-kh}{\pi h^2}$. (ii) Given that the depth stops increasing when $h=10$, find the value of $k$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Water drains from a cylindrical tank of cross-sectional area $4$ m$^2$. At time $t$ minutes the depth of water is $h$ m, and water leaves through a hole in the base at a rate of $k\sqrt{h}$ m$^3$ per minute, where $k$ is a positive constant.

   (a)

   (i) Show that $\frac{dh}{dt}=-\frac{k}{4}\sqrt{h}$. (ii) Given that the depth is decreasing at $0.15$ m per minute when $h=16$, find the value of $k$ and the rate at which the volume of water is decreasing when $h=9$.

   [6 marks]

   (b)

   Water is now also pumped into the tank at a constant rate of $0.6$ m$^3$ per minute, and the water still leaves through the hole with $k=0.15$. (i) Form a differential equation for $\frac{dh}{dt}$ in terms of $h$. (ii) Show that the depth is constant when $h=16$. (iii) Determine whether the depth is increasing or decreasing when $h=9$, and state the rate.

   [6 marks]

   Total for question 4: 12 marks

End of questions