Applying linear graphs is essential for plumbing professionals in Kenya to interpret relationships between variables such as water pressure, flow rates, and pipe lengths. Mastery of graphing helps in planning installations, estimating costs, and troubleshooting system performance. This chapter focuses on identifying relevant data from plumbing problems and choosing appropriate scales to create accurate and useful linear graphs.

7.1 Identifying Information

Plumbers often encounter complex data sets from site measurements, equipment specifications, and operational tests. Extracting the correct information needed for graphing ensures that the resulting graph accurately represents the plumbing system behavior. This section covers how to understand given data and select the relevant variables for plotting linear graphs.

7.1.1 Understanding given data sets and problems

Linear graphs represent relationships between two variables, typically an independent variable \(x\) and a dependent variable \(y\). Given data sets usually include multiple pairs of values that show how one variable changes with the other. Understanding the problem involves recognizing what each variable represents and the units involved.

The general linear relationship is expressed as:

$$y = mx + c$$

where \(m\) is the slope and \(c\) is the y-intercept.

Worked Examples

Example 1: A plumber measures water flow rate \(Q\) (litres per minute) at different pipe lengths \(L\) (metres). The data shows \(Q\) decreases linearly as \(L\) increases. Given the equation \(Q = -0.5L + 20\), find the flow rate at \(L = 10\) m.

Given: \(m = -0.5\), \(c = 20\), \(L = 10\) m$$Q = mL + c$$$$Q = (-0.5)(10) + 20$$$$Q = -5 + 20$$ Answer: 15 litres per minute

Example 2: Pressure loss \(P\) (kPa) in a pipe increases with flow velocity \(V\) (m/s) following \(P = 4V + 2\). Calculate \(P\) when \(V = 3\) m/s.

Given: \(m = 4\), \(c = 2\), \(V = 3\) m/s$$P = mV + c$$$$P = 4 \times 3 + 2$$$$P = 12 + 2$$ Answer: 14 kPa

Example 3: A leak detection device shows the relationship between time \(t\) (hours) and volume lost \(V\) (litres) as \(V = 1.2t + 0.5\). Find the volume lost after 8 hours.

Given: \(m = 1.2\), \(c = 0.5\), \(t = 8\) hours$$V = mt + c$$$$V = 1.2 \times 8 + 0.5$$$$V = 9.6 + 0.5$$ Answer: 10.1 litres

Example 4: The temperature \(T\) (°C) in a water tank increases linearly with heating time \(h\) (minutes) as \(T = 0.8h + 25\). Calculate the temperature after 45 minutes.

Given: \(m = 0.8\), \(c = 25\), \(h = 45\) minutes$$T = mh + c$$$$T = 0.8 \times 45 + 25$$$$T = 36 + 25$$ Answer: 61 °C

Example 5: The cost \(C\) (Ksh) of pipe material varies linearly with pipe length \(L\) (metres) as \(C = 250L + 500\). Find the cost for 15 metres of pipe.

Given: \(m = 250\), \(c = 500\), \(L = 15\) m$$C = mL + c$$$$C = 250 \times 15 + 500$$$$C = 3750 + 500$$ Answer: 4250 Ksh

7.1.2 Extracting relevant information for graphing

To plot a linear graph, it is important to identify the two variables to be graphed and their corresponding data points. In plumbing contexts, these variables often include flow rate, pressure, pipe length, or cost. Extracting relevant information involves selecting data that clearly shows a linear trend and discarding irrelevant or inconsistent data.

Worked Examples

Example 1: A plumber has the following data for water pressure \(P\) (kPa) and pipe length \(L\) (m):

Extract the variables suitable for graphing and describe the relationship.

Given: \(L\) and \(P\) values as above
Variables: \(x = L\), \(y = P\)
Relationship: \(P\) decreases linearly as \(L\) increases.

Example 2: Given data on flow velocity \(V\) (m/s) and flow rate \(Q\) (l/s):

Identify the dependent and independent variables for graphing.

Given: Data as above
Independent variable \(x = V\)
Dependent variable \(y = Q\)
Linear relationship \(Q\) increases as \(V\) increases.

Example 3: A data set includes water temperature \(T\) (°C), heating time \(h\) (min), and cost \(C\) (Ksh):

Which pair of variables is appropriate for a linear graph of temperature change?

Given: Variables \(h\), \(T\), and \(C\)
Appropriate pair: \(h\) (independent) and \(T\) (dependent)
Because heating time affects temperature linearly.

Example 4: You have data on pipe diameter \(d\) (mm) and flow rate \(Q\) (l/s):

Determine the variables to graph and whether the relation is linear.

Given: Data as above
Variables: \(x = d\), \(y = Q\)
Relationship: \(Q\) increases approximately linearly with \(d\).

Example 5: A plumber records water pressure \(P\) (kPa) at different flow rates \(Q\) (l/min):

Select variables for graphing and describe the trend.

Given: Data as above
Variables: \(x = Q\), \(y = P\)
Trend: Pressure decreases linearly as flow rate increases.

Chapter Summary

This chapter focused on applying linear graphs by first emphasizing the importance of identifying and understanding given data sets and extracting the relevant information needed for accurate graphing. It then covered how to choose appropriate scales for both axes, highlighting how scale selection affects the clarity and interpretation of the graph. Proper labeling of axes was explained as essential for clear communication of the variables and units involved, ensuring that the graph conveys meaningful information. The chapter detailed techniques for plotting points accurately on the graph and connecting them to form linear lines, which represent relationships between variables. Finally, it explored how to analyze the resulting graphs to identify trends and patterns, and how to draw valid conclusions from the data presented. This comprehensive approach equips students with the skills to construct and interpret linear graphs effectively in practical civil engineering contexts.