Problem 88 - Elementary and Intermediate Algebra

Question

Weather Forecasting. \(\quad\) A high-pressure area promises increasingly warmer weather for the next 48 hours. The temperature is now \(34^{\circ}\) Celsius and is expected to rise \(1^{\circ}\) every 6 hours. Write a composition function that expresses the Fahrenheit temperature as a function of the number of hours from now. (Hint: \(\left.F(C)=\frac{9}{5} C+32 .\right)\)

Step-by-Step Solution

Answer

The composition function is \( F(h) = \frac{3}{10}h + 93.2 \).

Step 1: Understand the problem

We have to find the temperature in Fahrenheit as a function of time in hours. We start with a current temperature of 34°C, which increases by 1°C every 6 hours.

Step 2: Define the function for temperature in Celsius

Let the number of hours from now be represented by the variable \( h \). The temperature increase is \( \frac{1}{6} \) degrees per hour. The Celsius temperature \( C(h) \) can be expressed as:\[ C(h) = 34 + \frac{1}{6} h \]

Step 3: Use the conversion formula to express temperature in Fahrenheit

Convert the temperature from Celsius to Fahrenheit using the conversion formula \( F(C) = \frac{9}{5}C + 32 \).

Step 4: Compose the functions to express Fahrenheit in terms of hours

Substitute \( C(h) \) into \( F(C) \) to get \( F(h) \):\[ F(h) = \frac{9}{5}(34 + \frac{1}{6}h) + 32 \]Simplify the expression:\[ F(h) = \frac{9}{5} \times 34 + \frac{9}{5} \times \frac{1}{6} h + 32 \]\[ F(h) = 61.2 + \frac{3}{10} h + 32 \]\[ F(h) = \frac{3}{10} h + 93.2 \]

Step 5: Present the function

Thus, the Fahrenheit temperature as a function of time in hours is given by:\[ F(h) = \frac{3}{10}h + 93.2 \]

Key Concepts

Celsius to Fahrenheit conversion

Understanding how to convert temperatures from Celsius to Fahrenheit is an important skill, especially in contexts where both temperature scales are used. The conversion formula is straightforward:

• Fahrenheit (F) = \( \frac{9}{5} \times \) Celsius (C) + 32

This formula basically scales the Celsius temperature by \( \frac{9}{5} \) and then adjusts for the starting point of the Fahrenheit scale with an addition of 32.

For example, if you have a temperature of \(34^{\circ}\) Celsius and want to find the equivalent in Fahrenheit, plug the Celsius value into the formula:

• F = \( \frac{9}{5} \times 34 + 32 = 61.2 + 32 \)
• F = 93.2

This means that \(34^{\circ}\) Celsius equals \(93.2^{\circ}\) Fahrenheit.

Using this conversion becomes especially useful when dealing with functions and trying to express a temperature in different units, like transforming a temperature function expressed in Celsius into one in Fahrenheit.

Linear function

A linear function is a mathematical expression that models a continuous constant change. Linear functions feature both input and output variables related by a constant rate of change, which is depicted as a straight line on a graph.

Linear functions can be represented by the equation:

• y = mx + b

Where:

• y is the dependent variable
• m is the slope of the line, representing the rate of change
• x is the independent variable
• b is the y-intercept, indicating the point at which the line crosses the y-axis

An example from our weather scenario looks like this:

• In Celsius, this is represented with: \(C(h) = 34 + \frac{1}{6}h\)

The function states that temperature increases by \(\frac{1}{6}\) degrees per hour. This fractional increase is our 'm' (slope), and 34 is our 'b' (y-intercept), the initial temperature.

Understanding linear functions is essential in everyday life because they can model various real-world phenomena, such as temperature changes or financial growth.

Temperature increase in Celsius

When predicting how temperature changes over time, the rate of increase or decrease is vital.

From the exercise, the temperature starts at \(34^{\circ}\) Celsius and is expected to rise by \(1^{\circ}\) Celsius every 6 hours. This situation can be modeled with a linear function, as the change is constant per unit of time. Given the parameter:

• Rate of increase per hour: \(\frac{1}{6}\) degrees

We derive the function:

• \(C(h) = 34 + \frac{1}{6}h\)

In this equation, \(C(h)\) is the Celsius temperature depending on time, h hours later. As time progresses, the temperature steadily increases by the specified rate.

This modeling method aligns with linear functions, reinforcing that the structure and consistency of the rate determine how effectively future values (temperatures in this case) are predicted. Understanding these increments helps with planning, especially in weather forecasts or climate studies.