Practical applications of linear functions

Modelling a real-life situation

Linear functions are used to model many situations in which one quantity varies in proportion to another, starting from a fixed initial value.

Example 1 – Car hire: a hire company charges a fixed fee of 30 euros plus 0.20 euros per kilometre travelled. The total cost for x km is:

C(x) = 0.20*x + 30

For 150 km, C(150) = 0.20 × 150 + 30 = 30 + 30 = 60 euros.

Example 2 – Temperature conversion: the formula for converting degrees Celsius (x) to Fahrenheit is:

F(x) = (9/5)*x + 32

For x = 20 degrees C, F(20) = 1.8*20 + 32 = 36 + 32 = 68 degrees F.

Comparing two offers

Example: Plan A costs 20 euros per month plus 0.05 euros per text message; Plan B costs 35 euros per month with no additional charges. We want to work out at what number of text messages Plan B becomes the better option.

A(x) = 0.05*x + 20 and B(x) = 35.

We solve A(x) > B(x): 0.05x + 20 > 35, so 0.05x > 15, so x > 300. Beyond 300 text messages, Plan B is better value.

Common pitfall

In a real-world problem, always check the units and the valid range (for example, a number of kilometres or text messages cannot be negative). A mathematically correct solution may not make sense in the context.