Chapter 4

The area \(A\) of a triangle is given by \(A=\frac{1}{2} b h\). The base \(b\) when measured is found to be \(3.26 \mathrm{~cm}\), and the perpendicular height \(h\) is \(7.5 \mathrm{~cm}\). Determine the area of the triangle.

State which type of error has been made in the following statements: (a) \(72 \times 31.429=2262.9\) (b) \(16 \times 0.08 \times 7=89.6\) (c) \(11.714 \times 0.0088=0.3247\), correct to 4 decimal places. (d) \(\frac{29.74 \times 0.0512}{11.89}=0.12\), correct to 2 significant figures.

Without using a calculator, determine an approximate value of (a) \(\frac{11.7 \times 19.1}{9.3 \times 5.7}\) (b) \(\frac{2.19 \times 203.6 \times 17.91}{12.1 \times 8.76}\)

Evaluate the following, correct to 4 significant figures: (a) \(4.7826+0.02713\) (b) \(17.6941-11.8762\) (c) \(21.93 \times 0.012981\)

Evaluate the following, correct to 4 decimal places: (a) \(46.32 \times 97.17 \times 0.01258\) (b) \(\frac{4.621}{23.76}\) (c) \(\frac{1}{2}(62.49 \times 0.0172)\)

Evaluate the following, correct to 3 decimal places: (a) \(\frac{1}{52.73}\) (b) \(\frac{1}{0.0275}\) (c) \(\frac{1}{4.92}+\frac{1}{1.97}\)

Evaluate the following, expressing the answers in standard form, correct to 4 significant figures. (a) \((0.00451)^{2}\) (b) \(631.7-(6.21+2.95)^{2}\) (c) \(46.27^{2}-31.79^{2}\)

Evaluate the following, correct to 3 decimal places: (a) \(\frac{(2.37)^{2}}{0.0526}\) (b) \(\left(\frac{3.60}{1.92}\right)^{2}+\left(\frac{5.40}{2.45}\right)^{2}\) (c) \(\frac{15}{7.6^{2}-4.8^{2}}\)

Evaluate the following, correct to 3 decimal places: (a) \(\frac{(2.37)^{2}}{0.0526}\) (b) \(\left(\frac{3.60}{1.92}\right)^{2}+\left(\frac{5.40}{2.45}\right)^{2}\) (c) \(\frac{15}{7.6^{2}-4.8^{2}}\)

Evaluate the following, correct to 3 decimal places: (a) \(\sqrt{0.007328}\) (b) \(\sqrt{52.91}-\sqrt{31.76}\) (c) \(\sqrt{\left(1.6291 \times 10^{4}\right)}\)

Evaluate the following, correct to 4 significant figures: (a) \(4.72^{3}\) (b) \((0.8316)^{4}\) (c) \(\sqrt{\left(76.21^{2}-29.10^{2}\right)}\)

Evaluate the following, correct to 3 significant figures: (a) \(\sqrt{\left[\frac{6.09^{2}}{25.2 \times \sqrt{7}}\right]}\) (b) \(\sqrt[3]{47.291}\) (c) \(\sqrt{\left(7.213^{2}+6.418^{3}+3.291^{4}\right)}\)

Evaluate the following, expressing the answers in standard form, correct to 4 decimal places: (a) \(\left(5.176 \times 10^{-3}\right)^{2}\) (b) \(\left(\frac{1.974 \times 10^{1} \times 8.61 \times 10^{-2}}{3.462}\right)^{4}\) (c) \(\left.\sqrt{\left(1.792 \times 10^{-4}\right.}\right)\)

Currency exchange rates for five countries are shown in Table \(4.1\) $$ \begin{aligned} &\text { Table } 4.1\\\ &\begin{array}{ll} \hline \text { France } & £ 1=1.50 \text { euros } \\ \text { Japan } & £ 1=175 \text { yen } \\ \text { Norway } & £ 1=11.25 \text { kronor } \\ \text { Switzerland } & £ 1=2.20 \text { francs } \\ \text { U.S.A. } & £ 1=1.82 \text { dollars (\$) } \\ \hline \end{array} \end{aligned} $$ Calculate: (a) how many French euros \(£ 27.80\) will buy, (b) the number of Japanese yen which can be bought for \(£ 23\), (c) the pounds sterling which can be exchanged for \(6412.5\) Norwegian kronor, (d) the number of American dollars which can be purchased for \(£ 90\), and (e) the pounds sterling which can be exchanged for 2794 Swiss francs.

Some approximate imperial to metric conversions are shown in Table \(4.2\) $$ \begin{aligned} &\text { Table } 4.2\\\ &\begin{array}{ll} \hline \text { length } & 1 \text { inch }=2.54 \mathrm{~cm} \\ & 1 \text { mile }=1.61 \mathrm{~km} \\ \text { weight } & 2.2 \mathrm{lb}=1 \mathrm{~kg} \\ & (1 \mathrm{lb}=16 \mathrm{oz}) \\ \text { capacity } & 1.76 \text { pints }=1 \text { litre } \\ & (8 \text { pints }=1 \text { gallon }) \\ \hline \end{array} \end{aligned} $$ Use the table to determine: (a) the number of millimetres in \(9.5\) inches, (b) a speed of 50 miles per hour in kilometres per hour, (c) the number of miles in \(300 \mathrm{~km}\), (d) the number of kilograms in 30 pounds weight, (e) the number of pounds and ounces in 42 kilograms (correct to the nearest ounce), (f) the number of litres in 15 gallons, and (g) the number of gallons in 40 litres.

In an electrical circuit the voltage \(V\) is given by Ohm's law, i.e. \(V=I R\). Find, correct to 4 significant figures, the voltage when \(I=5.36 \mathrm{~A}\) and \(R=14.76 \Omega\).

The surface area \(A\) of a hollow cone is given by \(A=\pi r l\). Determine, correct to 1 decimal place, the surface area when \(r=3.0 \mathrm{~cm}\) and \(l=8.5 \mathrm{~cm}\).

Velocity \(v\) is given by \(v=u+a t\). If \(u=9.86 \mathrm{~m} / \mathrm{s}, a=4.25 \mathrm{~m} / \mathrm{s}^{2}\) and \(t=6.84 \mathrm{~s}\), find \(v\), correct to 3 significant figures.

The area, \(A\), of a circle is given by \(A=\pi r^{2}\). Determine the area correct to 2 decimal places, given radius \(r=5.23 \mathrm{~m}\)

The power \(P\) watts dissipated in an electrical circuit may be expressed by the formula \(P=\frac{V^{2}}{R}\). Evaluate the power, correct to 3 significant figures, given that \(V=17.48 \mathrm{~V}\) and \(R=36.12 \Omega\).

The volume \(V \mathrm{~cm}^{3}\) of a right circular cone is given by \(V=\frac{1}{3} \pi r^{2} h\). Given that \(r=4.321 \mathrm{~cm}\) and \(h=\) \(18.35 \mathrm{~cm}\), find the volume, correct to 4 significant figures.

Force \(F\) newtons is given by the formula \(F=\frac{G m_{1} m_{2}}{d^{2}}\), where \(m_{1}\) and \(m_{2}\) are masses, \(d\) their distance apart and \(G\) is a constant. Find the value of the force given that \(G=6.67 \times 10^{-11}, m_{1}=7.36, m_{2}=15.5\) and \(d=22.6\). Express the answer in standard form, correct to 3 significant figures.

The time of swing \(t\) seconds, of a simple pendulum is given by \(t=2 \pi \sqrt{\frac{l}{g}}\). Determine the time, correct to 3 decimal places, given that \(l=12.0\) and \(g=9.81\).

Resistance, \(R \Omega\), varies with temperature according to the formula \(R=R_{0}(1+\alpha t)\). Evaluate \(R\), correct to 3 significant figures, given \(R_{0}=14.59, \alpha=0.0043\) and \(t=80\).