2.10 Logarithmic scaling and linearizing dataIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A researcher studies samples in which the number of cells $N$ ranges from 20 to 3 000 000, so she records $\log_{10}N$ instead of $N$.

   (a)

   Find $\log_{10}N$ when $N=3\,000\,000$.

   [1 mark]

   • A$6.00$
   • B$18.0$
   • C$0.477$
   • D$6.48$

   (b)

   On this scale, an increase of 1 in $\log_{10}N$ means that $N$ has been...

   [1 mark]

   • Aincreased by 10
   • Bincreased by 1
   • Cmultiplied by 10
   • Dmultiplied by $e$

   (c)

   A sample has $\log_{10}N=4.5$. Find $N$, correct to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of bacteria $N$ in a culture after $t$ hours is modelled by $N=ab^{t}$, where $a$ and $b$ are constants. The relationship between $\ln N$ and $t$ is linear, with gradient $0.35$ and vertical intercept $4.2$.

   (a)

   Which statement is correct?

   [1 mark]

   • A$\ln b=4.2$
   • B$\ln a=4.2$
   • C$a=0.35$
   • D$\ln a=0.35$

   (b)

   Find the value of $b$.

   [1 mark]

   • A$1.42$
   • B$0.35$
   • C$66.7$
   • D$-1.05$

   (c)

   Find the value of $a$ and state what it represents.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The orbital period $T$ (days) of a planet and its mean distance $d$ (million km) from the Sun are modelled by $T=kd^{n}$. Data for four planets, with $d$ from 57.9 to 228, give the line of best fit $\log_{10}T=1.5\log_{10}d-0.70$.

   (a)

   By taking logarithms of $T=kd^{n}$, find the values of $n$ and $k$.

   [3 marks]

   (b)

   Jupiter has a mean distance of 778 million km. Use the model to estimate its orbital period (i) in days, (ii) in years, taking 1 year as 365 days. (iii) Comment on the reliability of your estimate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The total number of cases $C$ of an illness in a city is recorded $t$ days after the first report. The results are: $t=2$, $C=23$; $t=4$, $C=66$; $t=6$, $C=190$; $t=8$, $C=540$; $t=10$, $C=1550$. A student suspects a model of the form $C=ab^{t}$, where $a>0$ and $b>1$.

   (a)

   (i) Show that, if $C=ab^{t}$, then $\ln C$ is a linear function of $t$.
   (ii) Use your GDC to find the equation of the regression line of $\ln C$ on $t$, and the value of Pearson's product-moment correlation coefficient $r$.
   (iii) Hence find the values of $a$ and $b$.

   [6 marks]

   (b)

   (i) Use your regression line to estimate the number of cases after 12 days.
   (ii) Find the time at which the model first predicts more than 10 000 cases.
   (iii) Comment on the validity of the model and of your answers to (i) and (ii).

   [6 marks]

   Total for question 4: 12 marks

End of questions