SPS SPS SM Pure 2023 June — Question 15 6 marks

Exam BoardSPS
ModuleSPS SM Pure (SPS SM Pure)
Year2023
SessionJune
Marks6
TopicLaws of Logarithms
TypeModel y=ax^b: linearise and find constants from graph/data
DifficultyModerate -0.5 This is a straightforward logarithmic transformation question requiring basic log laws (part a), finding gradient and intercept from two points (part b), and simple interpretation (part c). All techniques are standard A-level material with no novel problem-solving required, making it slightly easier than average.
Spec1.06f Laws of logarithms: addition, subtraction, power rules1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form2.02c Scatter diagrams and regression lines
The resting metabolic rate, \(R\) ml of oxygen consumed per hour, of a particular species of mammal is modelled by the formula, $$R = aM^b$$ where • \(M\) grams is the mass of the mammal • \(a\) and \(b\) are constants
  1. Show that this relationship can be written in the form $$\log_{10} R = b \log_{10} M + \log_{10} a$$ [2] \includegraphics{figure_3} A student gathers data for \(R\) and \(M\) and plots a graph of \(\log_{10} R\) against \(\log_{10} M\) The graph is a straight line passing through points \((0.7, 1.2)\) and \((1.8, 1.9)\) as shown in Figure 3.
  2. Using this information, find a complete equation for the model. Write your answer in the form $$R = aM^b$$ giving the value of each of \(a\) and \(b\) to 3 significant figures. [3]
  3. With reference to the model, interpret the value of the constant \(a\) [1]
The resting metabolic rate, $R$ ml of oxygen consumed per hour, of a particular species of mammal is modelled by the formula,

$$R = aM^b$$

where

• $M$ grams is the mass of the mammal

• $a$ and $b$ are constants

(a) Show that this relationship can be written in the form

$$\log_{10} R = b \log_{10} M + \log_{10} a$$ [2]

\includegraphics{figure_3}

A student gathers data for $R$ and $M$ and plots a graph of $\log_{10} R$ against $\log_{10} M$

The graph is a straight line passing through points $(0.7, 1.2)$ and $(1.8, 1.9)$ as shown in Figure 3.

(b) Using this information, find a complete equation for the model.
Write your answer in the form

$$R = aM^b$$

giving the value of each of $a$ and $b$ to 3 significant figures. [3]

(c) With reference to the model, interpret the value of the constant $a$ [1]

\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q15 [6]}}
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