Question 11:
Table 2 shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation xa = by, such that a and b are constants.
(a) Based on Table 2, construct a table for the values of log10x and log10y. [2 marks]
(b) Plot log10y against log10x, using a scale of 2 cm to 0.2 unit on both axes.
Hence, draw the line of best fit.
[3 marks]
(c) Write xa = by in linear form.
Hence, by using the graph in (b), find the value of a and of b.
[5 marks]
Answer:
(a)
(b)
(c)
$$ \begin{aligned} x^a & =b y \\ \log _{10} x^a & =\log _{10} b y \\ a \log _{10} x & =\log _{10} b+\log _{10} y \\ \log _{10} y & =a \log _{10} x-\log _{10} b \\ Y & =m \quad X+C \end{aligned} $$
$$ \begin{aligned} m=a & =\frac{0.8-0.2}{0.9-0.5} \\ a & =\frac{3}{2} \end{aligned} $$
$$ \begin{aligned} -\log _{10} b & =-0.54 \\ \log _{10} b & =0.54 \\ b & =10^{0.54} \\ b & =3.4674 \end{aligned} $$
Table 2 shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation xa = by, such that a and b are constants.
(a) Based on Table 2, construct a table for the values of log10x and log10y. [2 marks](b) Plot log10y against log10x, using a scale of 2 cm to 0.2 unit on both axes.
Hence, draw the line of best fit.
[3 marks]
(c) Write xa = by in linear form.
Hence, by using the graph in (b), find the value of a and of b.
[5 marks]
Answer:
(a)
(b)
(c)
$$ \begin{aligned} x^a & =b y \\ \log _{10} x^a & =\log _{10} b y \\ a \log _{10} x & =\log _{10} b+\log _{10} y \\ \log _{10} y & =a \log _{10} x-\log _{10} b \\ Y & =m \quad X+C \end{aligned} $$
$$ \begin{aligned} m=a & =\frac{0.8-0.2}{0.9-0.5} \\ a & =\frac{3}{2} \end{aligned} $$
$$ \begin{aligned} -\log _{10} b & =-0.54 \\ \log _{10} b & =0.54 \\ b & =10^{0.54} \\ b & =3.4674 \end{aligned} $$