Chapter 25

In a triangle \(X Y Z, \angle X=51^{\circ}, \angle Y=67^{\circ}\) and \(Y Z=15.2 \mathrm{~cm}\). Solve the triangle and find its area.

A car starts from rest and its speed is measured every second for \(6 \mathrm{~s}\) : \(\begin{array}{lllllccc}\text { Time } t(\mathrm{~s}) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \text { Speed } v(\mathrm{~m} / \mathrm{s}) & 0 & 2.5 & 5.5 & 8.75 & 12.5 & 17.5 & 24.0\end{array}\) Determine the distance travelled in 6 seconds (i.e. the area under the \(v / t\) graph), by (a) the trapezoidal rule, (b) the mid-ordinate rule, and (c) Simpson's rule.

A river is \(15 \mathrm{~m}\) wide. Soundings of the depth are made at equal intervals of \(3 \mathrm{~m}\) across the river and are as shown below. \(\begin{array}{llllllll}\text { Depth }(\mathrm{m}) & 0 & 2.2 & 3.3 & 4.5 & 4.2 & 2.4 & 0\end{array}\) Calculate the cross-sectional area of the flow of water at this point using Simpson's rule.

The areas of seven horizontal cross sections of a water reservoir at intervals of \(10 \mathrm{~m}\) are: \(210, \quad 250, \quad 320, \quad 350, \quad 290, \quad 230, \quad 170 \mathrm{~m}^{2}\) Calculate the capacity of the reservoir in litres.

The power used in a manufacturing process during a 6 hour period is recorded at intervals of 1 hour as shown below. \(\begin{array}{lrrrrrrr}\text { Time (h) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\ \text { Power }(\mathrm{kW}) & 0 & 14 & 29 & 51 & 45 & 23 & 0\end{array}\) Plot a graph of power against time and, by using the midordinate rule, determine (a) the area under the curve and (b) the average value of the power.