Problem 95 - A Graphical Approach to College Algebra

Question

Climate Change During the past 50 years, the average rate of change in temperature in Antarctica has been \(0.9^{\circ} \mathrm{F}\) per decade. (a) Write a function \(W\) that calculates the increase in temperature after \(x\) years during this time period. (b) Evaluate \(W(15)\) and interpret the result.

Step-by-Step Solution

Answer

The temperature increased by 1.35°F in 15 years.

Step 1: Understand the Rate of Change

The problem states that the temperature change rate in Antarctica is 0.9°F per decade. This means every 10 years the temperature rises by 0.9°F.

Step 2: Express the Rate Per Year

Convert the rate per decade into a rate per year. Since 1 decade is 10 years, the rate per year is \( \frac{0.9}{10} = 0.09^{\circ} \mathrm{F} \) per year.

Step 3: Define the Function W

The function \( W(x) \) calculates the increase in temperature after \( x \) years. Since the temperature increases by 0.09°F per year, the function is defined as \( W(x) = 0.09x \).

Step 4: Evaluate W(15)

Substitute \( x = 15 \) into the function \( W(x) \). Calculate \( W(15) = 0.09 \times 15 = 1.35 \).

Step 5: Interpret the Result

The result \( 1.35 \) means the temperature in Antarctica has increased by 1.35°F over the 15-year period.

Key Concepts

Linear Functions

A linear function is a type of mathematical relationship where the change between any two values of the function is constant. This means that if you were to graph the function, it would produce a straight line. Linear functions are often used in mathematical modeling to represent real-world situations because they can easily show relationships that change at a constant rate.

For this exercise, the linear function is used to model the change in temperature over time. The equation can be written in the form of \( W(x) = mx + b \), where \( m \) is the slope or rate of change, and \( b \) is the y-intercept, which is the starting value. In the problem, since we start counting from when the temperature change begins, \( b = 0 \). Therefore, the function simplifies to \( W(x) = 0.09x \).

To summarize, the linear function here models a consistent and predictable increase in temperature over time in Antarctica.

Rate of Change

The rate of change is a key concept in understanding linear functions. It describes how one quantity changes in relation to another. It can be seen as the slope of the line in a graph of a linear function.

In this exercise, the original rate of change is given as 0.9°F per decade for Antarctica. However, since we need to express this change per year for the function, we divide by 10. This gives us a rate of 0.09°F per year.

The rate of change can help in predicting how much things will change over time. In terms of climate change, understanding the rate at which temperature changes can help in forecasting future conditions and developing strategies to address these changes.

Function Evaluation

Function evaluation involves calculating the output of a function for a particular input. This is where the defined function comes into play to give meaningful predictions or insights.

In our example, the function \( W(x) = 0.09x \) is evaluated by substituting the number of years, \( x \), into the equation. When asked to evaluate \( W(15) \), you substitute 15 for \( x \), calculate \( W(15) = 0.09 \times 15 = 1.35 \).

This tells us that after 15 years, the temperature in Antarctica has increased by 1.35°F as modeled by the function. Function evaluation is useful because it turns mathematical expressions into real-world predictions, making it easier to understand and apply in various situations.