Can You Provide a Real-Life Problem Involving a Linear Function?

Linear functions are ubiquitous in day-to-day life and are used to describe processes with a constant rate. They can be seen in various contexts such as taxi fares, raffle ticket pricing, and conversions between different units. This article explores a few real-life problems involving linear functions, including the pricing of taxi rides and raffle tickets, and the conversions between different units.

Example 1: Tom's Taxi Service

Tom's Taxi Service charges a base fee of $5, plus an additional $2 per mile driven. This can be represented by a linear function.

Solution

Let's break it down:

Base fee: $5 (constant term) Additional cost per mile: $2 (slope)

Linear function:

Cx  2x   5

Where:

Cx: Total cost (in dollars) x: Number of miles driven 2: Slope (additional cost per mile) 5: Y-intercept (base fee)

Example Usage

Find the total cost for a 10-mile trip:

C10  2 * 10   5C10  20   5C10  25

So the total cost for a 10-mile trip is $25.

Example 2: Raffle Tickets at a Carnival

A carnival barker is selling tickets to a raffle for the chance to win a Grand Prize. The first ticket is priced at $5, and each additional ticket is $3.

Problem a: Clarissa Purchased 20 Tickets

Find out how much Clarissa paid for her 20 tickets:

Cx  5   3(x - 1)C20  5   3(20 - 1)C20  5   3 * 19C20  5   57C20  62

So, Clarissa paid $62 for 20 tickets.

Problem b: Althea Spent $116 on Raffle Tickets

Find out how many tickets Althea purchased:

Cx  5   3(x - 1)116  5   3(x - 1)116 - 5  3(x - 1)111  3(x - 1)37  x - 1x  38

So, Althea purchased 38 tickets.

Applications of Linear Functions in Everyday Life

Linear functions describe any process with a constant rate, such as a constant speed, constant consumption, or constant time passing.

If you drive at an average speed of 'v' how far will you have gone after time 't': Distance Speed * Time (d vt) If you're driving at an average 'v' speed, how long will it take to cover a distance 'd': Time Distance / Speed (t d / v) If you drive a distance 'd' in a given time 't', what was your average speed: Speed Distance / Time (v d / t)

These linear functions allow us to make forecasts and predictions in various scenarios, such as driving times, shopping for food, or even setting up billing structures for services like Tom's Taxi Service or raffle ticket pricing.

Conversions of Units

Linear functions are also used in unit conversions. Here are some common conversions:

Temperature Conversions: Between degrees Fahrenheit and degrees Celsius, Kelvin, and degrees Fahrenheit, and Fahrenheit and Rankine. Units of Power: Between Watts and Horsepower, and between British Thermal Units (BTU) and Calories. Units of Distance: Between Miles and Light-years. Units of Currency: Between Pounds Sterling and Australian Dollars.

Each of these conversions involves a linear function. For example, the conversion between degrees Fahrenheit (F) and degrees Celsius (C) is given by:

C  (5/9) * (F - 32)

These conversions are linear and can be represented by the form y mx b, where m is the conversion factor and b is often 0 in the case of pure linear conversions.

In conclusion, linear functions are pervasive in our daily lives, playing a crucial role in predicting and describing various real-world phenomena and conversions. By understanding and applying these functions, we can make accurate forecasts and solve practical problems efficiently.