IB Questionbank

User interface language: English | Español

Date	May 2021	Marks available	6	Reference code	21M.1.AHL.TZ1.12
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Show that	Question number	12	Adapted from	N/A

Question

A tank of water initially contains $400$ litres. Water is leaking from the tank such that after $10$ minutes there are $324$ litres remaining in the tank.

The volume of water, $V$ litres, remaining in the tank after $t$ minutes, can be modelled by the differential equation

$\frac{d V}{d t} = - k \sqrt{V}$ , where $k$ is a constant.

Show that $V = {(20 - \frac{t}{5})}^{2}$ .

[6]

Find the time taken for the tank to empty.

[2]

Markscheme

$\frac{d V}{d t} = - k V^{\frac{1}{2}}$

use of separation of variables (M1)

$\Rightarrow \int V^{- \frac{1}{2}} d V = \int - k d t$ A1

$2 V^{\frac{1}{2}} = - k t (+ c)$ A1

considering initial conditions $40 = c$ A1

$2 \sqrt{324} = - 10 k + 40$

$\Rightarrow k = 0.4$ A1

$2 \sqrt{V} = - 0.4 t + 40$

$\Rightarrow \sqrt{V} = 20 - 0.2 t$ A1

Note: Award A1 for any correct intermediate step that leads to the AG.

$\Rightarrow V = {(20 - \frac{t}{5})}^{2}$ AG

Note: Do not award the final A1 if the AG line is not stated.

[6 marks]

$0 = {(20 - \frac{t}{5})}^{2} \Rightarrow t = 100$ minutes (M1)A1

[2 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » AHL 5.14—Setting up a DE, solve by separating variables

Show 26 related questions

22M.3.AHL.TZ1.1b.i:
Find the general solution of the differential equation $Q' (t) = β N Q (t)$ .
SPM.2.AHL.TZ0.7e:
By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.
SPM.2.AHL.TZ0.7g:
Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.
SPM.2.AHL.TZ0.7c:
Write down the differential equation as a system of first order differential equations.
SPM.2.AHL.TZ0.7f:
Use the differential equation to find the terminal velocity for the object.
SPM.2.AHL.TZ0.7b:
From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.
16N.2.AHL.TZ0.H_6:
An earth satellite moves in a path that can be described by the curve $72.5{x^2} + 71.5{y^2} = 1$ where $x = x(t)$ and $y = y(t)$ are in thousands of kilometres and $t$ is time in seconds.

Given that $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}$ when $x = 3.2 \times {10^{ - 3}}$ , find the possible values of $\frac{{{\text{d}}y}}{{{\text{d}}t}}$ .

Give your answers in standard form.
SPM.2.AHL.TZ0.7a:
By substituting $v = \frac{{{\text{d}}x}}{{{\text{d}}t}}$ into the equation, find an expression for the velocity of the particle at time $t$ . Give your answer in the form $v = f(t)$ .
21M.3.AHL.TZ2.2d.ii:
Show that the expression for the mackerel population density after $t$ years is $M = M_{0} e^{α t}$
21M.1.AHL.TZ1.12b:
Find the time taken for the tank to empty.
21M.3.AHL.TZ2.2d.i:
Write down the differential equation for $M$ that models this situation.
21M.3.AHL.TZ2.2d.iii:
Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that $α = 0.549$ to three significant figures.
SPM.2.AHL.TZ0.7d:
Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when $t = 0.6$ .
21M.2.AHL.TZ1.7a:
Find the population of rabbits $1$ year after they were introduced.
21N.3.AHL.TZ0.2a.i:
Find the equation of the regression line of $h$ on $t$ .
21N.3.AHL.TZ0.2a.ii:
Interpret the meaning of parameter $a$ in the context of the model.
21N.3.AHL.TZ0.2a.iii:
Suggest why Eva’s use of the linear regression equation in this way could be unreliable.
21N.3.AHL.TZ0.2b.i:
Find the equation of the least squares quadratic regression curve.
21N.3.AHL.TZ0.2b.ii:
Find the value of $k$ .
21N.3.AHL.TZ0.2b.iii:
Hence, write down a suitable domain for Eva’s function $h (t) = p t^{2} + q t + r$ .
21N.3.AHL.TZ0.2c:
Show that $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ .
21N.3.AHL.TZ0.2d:
By solving the differential equation $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ , show that the general solution is given by $h = 17 640 {(c - R^{2} t)}^{2}$ , where $c \in ℝ$ .
21N.3.AHL.TZ0.2e:
Use the general solution from part (d) and the initial condition $h (0) = 3.2$ to predict the value of $T$ .
21N.3.AHL.TZ0.2f:
Find this new height.
21N.3.AHL.TZ0.2g.i:
Show that $\frac{d H}{d t} \approx 0.2514 - 0.009873 t - 0.1405 \sqrt{H}$ , where $0 \leq t \leq T$ .
21N.3.AHL.TZ0.2g.ii:
Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$ .

Hide related questions

Topic 5—Calculus

Question

Markscheme

Examiners report

Syllabus sections

View options