Chapter 12

Plot the graph \(y=4 x+3\) in the range \(x=-3\), to \(x=+4\). From the graph, find (a) the value of \(y\) when \(x=2.2\), and (b) the value of \(x\) when \(y=-3\).

Plot the following graphs on the same axes between the range \(x=-4\) to \(x=+4\), and determine the gradient of each. (a) \(y=x\) (b) \(y=x+2\) (c) \(y=x+5\) (d) \(y=x-3\)

Plot the following graphs on the same axes between the values \(x=-3\) to \(x=+3\) and determine the gradient and \(y\)-axis intercept of each. (a) \(y=3 x\) (b) \(y=3 x+7\) (c) \(y=-4 x+4\) (d) \(y=-4 x-5\)

The following equations represent straight lines. Determine, without plotting graphs, the gradient and \(y\)-axis intercept for each. (a) \(y=3\) (b) \(y=2 x\) (c) \(y=5 x-1\) (d) \(2 x+3 y=3\)

Without plotting graphs, determine the gradient and \(y\)-axis intercept values of the following equations: (a) \(y=7 x-3\) (b) \(3 y=-6 x+2\) (c) \(y-2=4 x+9\) (d) \(\frac{y}{3}=\frac{x}{3}-\frac{1}{5}\) (e) \(2 x+9 y+1=0\)

Determine the gradient of the straight line graph passing through the co- ordinates (a) \((-2,5)\) and \((3,4)\), and (b) \((-2,-3)\) and \((-1,3)\)

Plot the graph \(3 x+y+1=0\) and \(2 y-5=x\) on the same axes and find their point of intersection.

The temperature in degrees Celsius and the corresponding values in degrees Fahrenheit are shown in the table below. Construct rectangular axes, choose a suitable scale and plot a graph of degrees Celsius (on the horizontal axis) against degrees Fahrenheit (on the vertical scale). $$ \begin{array}{|l|llrrrr|} \hline{ }^{\circ} \mathrm{C} & 10 & 20 & 40 & 60 & 80 & 100 \\ { }^{\circ} \mathrm{F} & 50 & 68 & 104 & 140 & 176 & 212 \\ \hline \end{array} $$ From the graph find (a) the temperature in degrees Fahrenheit at \(55^{\circ} \mathrm{C}\), (b) the temperature in degrees Celsius at \(167^{\circ} \mathrm{F}\), (c) the Fahrenheit temperature at \(0^{\circ} \mathrm{C}\), and (d) the Celsius temperature at \(230^{\circ} \mathrm{F}\).

In an experiment on Charles's law, the value of the volume of gas, \(V \mathrm{~m}^{3}\), was measured for various temperatures \(T^{\circ} \mathrm{C}\). Results are shown below. $$ \begin{array}{|l|llllll|} \hline V \mathrm{~m}^{3} & 25.0 & 25.8 & 26.6 & 27.4 & 28.2 & 29.0 \\ T^{\circ} \mathrm{C} & 60 & 65 & 70 & 75 & 80 & 85 \\ \hline \end{array} $$ Plot a graph of volume (vertical) against temperature (horizontal) and from it find (a) the temperature when the volume is \(28.6 \mathrm{~m}^{3}\), and (b) the volume when the temperature is \(67^{\circ} \mathrm{C}\).

In an experiment demonstrating Hooke's law, the strain in an aluminium wire was measured for various stresses. The results were: $$ \begin{aligned} &\begin{array}{|l|lll|} \hline \text { Stress } \mathrm{N} / \mathrm{mm}^{2} & 4.9 & 8.7 & 15.0 \\ \text { Strain } & 0.00007 & 0.00013 & 0.00021 \\ \hline \end{array}\\\ &\begin{array}{|l|ccc|} \hline \text { Stress N/mm }^{2} & 18.4 & 24.2 & 27.3 \\ \text { Strain } & 0.00027 & 0.00034 & 0.00039 \\ \hline \end{array} \end{aligned} $$ Plot a graph of stress (vertically) against strain (horizontally). Find: (a) Young's Modulus of Elasticity for aluminium which is given by the gradient of the graph, (b) the value of the strain at a stress of \(20 \mathrm{~N} / \mathrm{mm}^{2}\), and (c) the value of the stress when the strain is \(0.00020\)

The following values of resistance \(R\) ohms and corresponding voltage \(V\) volts are obtained from a test on a filament lamp. $$ \begin{array}{|l|lllrr|} \hline R \text { ohms } & 30 & 48.5 & 73 & 107 & 128 \\ V \text { volts } & 16 & 29 & 52 & 76 & 94 \\ \hline \end{array} $$ Choose suitable scales and plot a graph with \(R\) representing the vertical axis and \(V\) the horizontal axis. Determine (a) the gradient of the graph, (b) the \(R\) axis intercept value, (c) the equation of the graph, (d) the value of resistance when the voltage is \(60 \mathrm{~V}\), and (e) the value of the voltage when the resistance is \(40 \mathrm{ohms}\), (f) If the graph were to continue in the same manner, what value of resistance would be obtained at \(110 \mathrm{~V}\) ?

Experimental tests to determine the breaking stress \(\sigma\) of rolled copper at various temperatures \(t\) gave the following results $$ \begin{aligned} &\begin{array}{|l|ccc|} \hline \text { Stress } \sigma \mathrm{N} / \mathrm{cm}^{2} & 8.46 & 8.04 & 7.78 \\ \text { Temperature } t^{\circ} \mathrm{C} & 70 & 200 & 280 \\ \hline \end{array}\\\ &\begin{array}{|l|ccc|} \hline \text { Stress } \sigma \mathrm{N} / \mathrm{cm}^{2} & 7.37 & 7.08 & 6.63 \\ \text { Temperature } t^{\circ} \mathrm{C} & 410 & 500 & 640 \\ \hline \end{array} \end{aligned} $$ Show that the values obey the law \(\sigma=a t+b\), where \(a\) and \(b\) are constants and determine approximate values for \(a\) and \(b\). Use the law to determine the stress at \(250^{\circ} \mathrm{C}\) and the temperature when the stress is \(7.54 \mathrm{~N} / \mathrm{cm}^{2}\).