Problem 88 - Intermediate Algebra

Question

Weather Forecasting. 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 function is \(F(h) = 93.2 + 0.3h\), where \(h\) is the number of hours.

Step 1: Understand the Problem

We are given a temperature in Celsius that starts at \(34^{\circ}\) Celsius and increases \(1^{\circ}\) every 6 hours for the next 48 hours. We need a function that converts this temperature to Fahrenheit and expresses it as a function of the number of hours that have passed.

Step 2: Define the Celsius Temperature Function

Define a function \(C(h)\) to describe the temperature in Celsius after \(h\) hours. Starting at \(34^{\circ}\) Celsius, the temperature increases by \(\frac{h}{6}\) degrees every 6 hours.\[ C(h) = 34 + \frac{h}{6} \]

Step 3: Define the Fahrenheit Conversion Function

Use the provided formula \(F(C) = \frac{9}{5}C + 32\) to convert Celsius to Fahrenheit. To express Fahrenheit as a function of hours \(h\), we apply this conversion to \(C(h)\).

Step 4: Compose the Functions

Substitute \(C(h)\) into the Fahrenheit formula:\[ F(h) = \frac{9}{5}(34 + \frac{h}{6}) + 32 \]This represents the temperature in Fahrenheit as a function of the time in hours.

Step 5: Simplify the Function

Simplify \(F(h)\):1. Distribute \(\frac{9}{5}\):\[ F(h) = \frac{9}{5} \times 34 + \frac{9}{5} \times \frac{h}{6} + 32 \]2. Simplify terms:\[ F(h) = \frac{306}{5} + \frac{3h}{10} + 32 \]3. Convert constants to a single fraction:\[ F(h) = \frac{306}{5} + \frac{320}{10} + \frac{3h}{10} \]4. Combine the terms:\[ F(h) = \frac{612}{10} + \frac{320}{10} + \frac{3h}{10} = \frac{932}{10} + \frac{3h}{10} \]5. Simplify:\[ F(h) = 93.2 + 0.3h \]

Key Concepts

Celsius to Fahrenheit conversion

Converting temperatures from Celsius to Fahrenheit is common in weather forecasting, especially if you are dealing with different temperature scales across regions. Understanding this conversion is crucial to navigate between the metric and imperial systems comfortably.
To convert Celsius to Fahrenheit, you can use the formula:

• \[ F(C) = \frac{9}{5} C + 32 \]

This formula helps you find the equivalent Fahrenheit temperature for a given Celsius temperature. The \( \frac{9}{5} \) factor accounts for the size difference between degrees in each scale, and the addition of 32 adjusts for the offset between the scales.
For example, if you have a temperature of 20°C, substituting into the formula gives:

• \[ F(20) = \frac{9}{5} \times 20 + 32 = 68^{\circ}F \]

This conversion is used not just in weather forecasts but in many scientific computations where temperature is a factor.

Linear functions

Linear functions are the backbone of understanding changes and predictions in various fields such as economics, science, and weather. These functions are expressed in the form \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.
In our weather scenario, the change in temperature over time can be represented as a linear function. When predicting temperature increases by a set amount over defined intervals, a linear function is a perfect fit.

• The Celsius temperature increases linearly over time, defined by \( C(h) = 34 + \frac{h}{6} \)

Here, the slope \( \frac{1}{6} \) tells us the temperature rises by 1°C every 6 hours, and the y-intercept 34°C is the starting temperature.
This simplifies understanding and making predictions as it translates real-world phenomena into equations that can be graphed, analyzed, and extrapolated easily.

Weather forecasting

Weather forecasting integrates complex systems of mathematics, physics, and computer science to predict atmospheric conditions. One vital aspect is understanding how temperature changes impact weather forecasts.
Temperature predictions involve linear functions, as seen in our example. Knowing how temperature will change over time helps meteorologists predict how weather patterns will evolve. For instance, as high-pressure systems move, they influence temperature increases or decreases.

• In the exercise, the temperature rise by 1°C every 6 hours forecasts warmer weather.
• The transition to Fahrenheit provides insights for regions using that scale while maintaining accurate predictions.

Unifying temperature scales ensures that forecasts meet local needs without losing the integrity of the scientific data. This contributes to more accurate and reliable weather predictions that help in planning and preparation for various activities or events.