IB Questionbank

User interface language: English | Español

Date	May Example question	Marks available	2	Reference code	EXM.2.AHL.TZ0.13
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Calculate	Question number	13	Adapted from	N/A

Question

Beth goes for a run. She uses a fitness app to record her distance, $s$ km, and time, $t$ minutes. A graph of her distance against time is shown.

Beth runs at a constant speed of 2.3 ms^–1 for the first 8 minutes.

Between 8 and 20 minutes, her distance can be modeled by a cubic function, $s = a{t^3} + b{t^2} + ct + d$ . She reads the following data from her app.

Hence find

Calculate her distance after 8 minutes. Give your answer in km, correct to 3 decimal places.

[2]

Find the value of $a$ , $b$ , $c$ and $d$ .

[5]

the distance she runs in 20 minutes.

[2]

c.i.

her maximum speed, in ms^–1.

[4]

c.ii.

Markscheme

$\frac{{2.3 \times 8 \times 60}}{{1000}} = 1.104$ M1A1

[2 marks]

either using a cubic regression or solving a system of 4 equations M1

$a = - 0.00364{\text{,}}\,\,b = 0.150{\text{,}}\,\,c = - 1.67{\text{,}}\,\,d = 6.72$ A1A1A1A1

[5 marks]

$s\left( {20} \right) = 4.21$ km (Note: Condone $s\left( {20} \right) = 4.2$ km obtained from using rounded values.) M1A1

[2 marks]

c.i.

EITHER finding maximum of $\frac{{ds}}{{dt}}$ OR solving $\frac{{{d^2}s}}{{d{t^2}}} = 0$ M1

maximum speed = 0.390… km per minute A1

maximum speed = 6.51 ms^–1 M1A1

[4 marks]

c.ii.

Examiners report

[N/A]

c.i.

[N/A]

c.ii.

Syllabus sections

Topic 2—Functions » AHL 2.9—HL modelling functions

Show 49 related questions

18M.2.SL.TZ2.S_6c:
Find when the seat is 30 m above the ground for the third time.
EXM.3.AHL.TZ0.9e:
Give two reasons why the prediction in part (b)(ii) might be lower than 14.
EXM.3.AHL.TZ0.9f.ii:
$c$ .
EXM.3.AHL.TZ0.9b.i:
the number of new people infected on day 6.
EXM.3.AHL.TZ0.9a:
Use an exponential regression to find the value of $a$ and of $b$ , correct to 4 decimal places.
22M.3.AHL.TZ1.1f.ii:
The solution to the differential equation is given by

$Q (t) = \frac{L}{1 + C e^{- k t}}$

where $C$ is a constant.

Using your answer to part (f)(i), estimate the percentage of computers in city X that are expected to have been infected by the virus over a long period of time.
22M.1.AHL.TZ1.17:
A function $f$ is of the form $f (t) = p e^{q \cos (r t)}, p, q, r \in ℝ^{+}$ . Part of the graph of $f$ is shown.

The points $A$ and $B$ have coordinates $A (0, 6.5)$ and $B (5.2, 0.2)$ , and lie on $f$ .

The point $A$ is a local maximum and the point $B$ is a local minimum.

Find the value of $p$ , of $q$ and of $r$ .
EXM.2.AHL.TZ0.13c.i:
the distance she runs in 20 minutes.
18M.2.SL.TZ1.S_10b.ii:
For the graph of $f$ , write down the period.
EXM.3.AHL.TZ0.9b.ii:
the day when the total number of people infected will be greater than 1000.
EXM.3.AHL.TZ0.9f.i:
$L$ .
EXM.3.AHL.TZ0.9f.iii:
$k$ .
EXM.3.AHL.TZ0.9c:
Use your answer to part (a) to show that the model predicts 16.7 people will be infected on the first day.
18M.2.SL.TZ1.S_10a:
Find the coordinates of A.
17M.2.SL.TZ1.S_8a.i:
How much time is there between the first low tide and the next high tide?
EXM.2.AHL.TZ0.13b:
Find the value of $a$ , $b$ , $c$ and $d$ .
EXM.3.AHL.TZ0.9h:
Use the logistic model to find the day when the rate of increase of people infected is greatest.
EXM.3.AHL.TZ0.9d.ii:
Perform a ${\chi ^2}$ goodness of fit test at the 5% significance level. You should clearly state your hypotheses, the p-value, and your conclusion.
EXM.3.AHL.TZ0.9d.i:
Explain why the number of degrees of freedom is 2.
17M.2.SL.TZ1.S_8c:
There are two high tides on 12 December 2017. At what time does the second high tide occur?
EXM.3.AHL.TZ0.9g:
Hence predict the total number of people infected by this disease after several months.
17M.2.SL.TZ1.S_8b.i:
Find the value of $p$ ;
18M.2.SL.TZ1.S_10d:
Find the maximum speed of the ball.
EXN.3.AHL.TZ0.1g.ii:
Use these parameters to calculate the value of $N (8)$ predicted by this model.
20N.2.SL.TZ0.S_8d:
Determine the rate of change of $h$ when the top of the bucket is at $B$ .
20N.2.SL.TZ0.S_8b:
Find $k$ .
18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
18M.2.SL.TZ1.S_10e:
Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².
21M.3.AHL.TZ1.1b:
Find the angle through which Mars rotates on its axis each hour.
20N.2.SL.TZ0.S_8a.i:
Find the height of point $A$ above the ground.
20N.2.SL.TZ0.S_8c:
Find $A \hat{O} B$ .
21M.3.AHL.TZ1.1a:
Show that $b \approx 0.00939$ .
17M.2.SL.TZ1.S_8b.iii:
Find the value of $r$ .
18M.2.SL.TZ1.S_10b.i:
For the graph of $f$ , write down the amplitude.
21M.3.AHL.TZ1.1c.ii:
Find the minimum value of $ω$ .
21M.3.AHL.TZ1.1d.ii:
the minimum value of $R (t)$ .
21M.3.AHL.TZ1.1g:
Find the value of $d$ .
EXM.2.AHL.TZ0.13c.ii:
her maximum speed, in ms^–1.
21M.3.AHL.TZ1.1e:
Hence show that $a = 1.6$ , correct to two significant figures.
20N.2.SL.TZ0.S_8a.iii:
Hence find the number of rotations the water wheel makes in one hour.
17M.2.SL.TZ1.S_8b.ii:
Find the value of $q$ ;
20N.2.SL.TZ0.S_8a.ii:
Calculate the number of seconds it takes for the water wheel to complete one rotation.
17M.2.SL.TZ1.S_8a.ii:
Find the difference in height between low tide and high tide.
21M.3.AHL.TZ1.1d.i:
the maximum value of $R (t)$ .
21M.3.AHL.TZ1.1c.i:
Show that the maximum value of $ω = 1.98$ , correct to three significant figures.
21M.3.AHL.TZ1.1f:
Find the value of $c$ .
EXN.1.AHL.TZ0.8b:
Write down a sequence of transformations that will transform the graph of $y = \cos x$ onto the graph of $y = f (x)$ .
EXN.3.AHL.TZ0.1g.i:
Assuming a carrying capacity of $5000$ use the given values of $N (0)$ and $N (1)$ to calculate the parameters $C$ and $k$ .
EXN.3.AHL.TZ0.1h:
Comment on the likelihood of the fish population reaching $5000$ .

Hide related questions

Topic 2—Functions

Question

Markscheme

Examiners report

Syllabus sections

View options