5.9 Differentiation rules and related ratesIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The depth of water, $h$ metres, in a harbour $t$ hours after midnight is modelled by $h(t)=6+2\sin(0.5t)$ for $0\le t\le 12$, where the angle is in radians.

   (a)

   Find $\dfrac{dh}{dt}$.

   [1 mark]

   • A$2\cos(0.5t)$
   • B$-\cos(0.5t)$
   • C$\cos(0.5t)$
   • D$0.5\cos(0.5t)$

   (b)

   Use your GDC, in radian mode, to find the rate of change of the depth at 04:00.

   [1 mark]

   • A$-0.416$ m h$^{-1}$
   • B$0.416$ m h$^{-1}$
   • C$0.999$ m h$^{-1}$
   • D$-0.909$ m h$^{-1}$

   (c)

   Find the first time after midnight at which the depth is greatest, and state the greatest depth.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The concentration of a drug in a patient's blood, $C$ mg l$^{-1}$, is modelled by $C(t)=\dfrac{20t}{t^2+4}$ for $t\ge0$, where $t$ is the time in hours after the drug is given.

   (a)

   Find $\dfrac{dC}{dt}$.

   [1 mark]

   • A$\dfrac{20(t^2-4)}{(t^2+4)^2}$
   • B$\dfrac{20}{2t}$
   • C$\dfrac{20(t^2+4)+40t^2}{(t^2+4)^2}$
   • D$\dfrac{20(4-t^2)}{(t^2+4)^2}$

   (b)

   Find the time at which the concentration is greatest.

   [1 mark]

   • A$4$ hours
   • B$2$ hours
   • C$5$ hours
   • D$0$ hours

   (c)

   Find the rate of change of the concentration at $t=1$ and state whether the concentration is increasing or decreasing.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The concentration of a chemical in a reaction vessel, $g$ mol dm$^{-3}$, is modelled by $g(t)=\sqrt{t}\,e^{-0.5t}$ for $t>0$, where $t$ is the time in minutes after mixing.

   (a)

   Find an expression for $\dfrac{dg}{dt}$.

   [3 marks]

   (b)

   Find the greatest concentration, justifying that it is a maximum.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A spherical balloon is inflated so that its volume, $V$ cm$^3$, increases at a constant rate of $50$ cm$^3$ s$^{-1}$. The radius of the balloon is $r$ cm. The volume of a sphere is $V=\frac43\pi r^3$ and its surface area is $A=4\pi r^2$.

   (a)

   (i) Find the rate of change of the radius when $r=10$.
   (ii) Find the rate of change of the surface area when $r=10$.

   [6 marks]

   (b)

   The balloon is empty when $t=0$, where $t$ is in seconds.
   (i) Find the radius when $t=20$.
   (ii) Show that $r=\left(\dfrac{150t}{4\pi}\right)^{1/3}$, and hence find $\dfrac{dr}{dt}$ when $t=20$.

   [6 marks]

   Total for question 4: 12 marks

End of questions