EXM.1.AHL.TZ0.14

pestleMathematicsAIHLPaper 1EXM· ahl-2-10-scaling-large-numbers-log-log-graphssource ↗

It is believed that two variables, $v$ and $w$ are related by the equation $v = k{w^n}$ , where $k{\text{,}}\,\,n \in \mathbb{R}$ . Experimental values of $v$ and $w$ are obtained. A graph of ${\text{ln}}\,v$ against ${\text{ln}}\,w$ shows a straight line passing through (−1.7, 4.3) and (7.1, 17.5).

Find the value of $k$ and of $n$ .

Markscheme / solution

${\text{ln}}\,v = n\,{\text{ln}}\,w + {\text{ln}}\,k$ M1A1

gradient $= \frac{{17.5 - 4.3}}{{7.1 + 1.7}}{\text{ }}\left( { = 1.5} \right)$ M1

$n = 1.5$ A1

$y$ -intercept $= 1.5 \times 1.7 + 4.3{\text{ }}\left( { = 6.85} \right)$ M1

$k = {e^{6.85}} = 944$ M1A1

[7 marks]

Examiners’ report

[N/A]