Problem 64 - Algebra for College Students

Question

Solve each problem by setting up and solving an appropriate inequality. The temperatures for a 24-hour period ranged between \(-4^{\circ} \mathrm{F}\) and \(23^{\circ} \mathrm{F}\), inclusive. What was the range in Celsius degrees? \(\left(\right.\) Use \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\).)

Step-by-Step Solution

Answer

The range in Celsius is 15 degrees.

Step 1: Convert the Fahrenheit Temps to Celsius

The formula for converting Fahrenheit to Celsius is given by \( F = \frac{9}{5}C + 32 \). First, let's convert the lowest temperature, \(-4^{\circ} \mathrm{F}\), to Celsius. Set \( F = -4 \) and solve for \( C \):\[ -4 = \frac{9}{5}C + 32 \]Subtract 32 from both sides:\[ -36 = \frac{9}{5}C \]Multiply both sides by \( \frac{5}{9} \):\[ C = -20 \]Now convert the highest temperature, \(23^{\circ} \mathrm{F}\), using the same method. Set \( F = 23 \) and solve for \( C \):\[ 23 = \frac{9}{5}C + 32 \]Subtract 32 from both sides:\[ -9 = \frac{9}{5}C \]Multiply both sides by \( \frac{5}{9} \):\[ C = -5 \]

Step 2: Determine the Temperature Range in Celsius

After converting the temperatures, we found that the range in Celsius is from \(-20^{\circ}\) to \(-5^{\circ}\). To find the range of temperatures in Celsius, subtract the lower bound from the upper bound:\[ -5 - (-20) = -5 + 20 = 15 \]

Key Concepts

Fahrenheit to Celsius conversion

Converting temperatures from Fahrenheit to Celsius is essential when dealing with weather data from countries that use different temperature scales. The conversion formula is \[ F = \frac{9}{5}C + 32 \]. This formula allows us to transition between the Fahrenheit scale, which is commonly used in the United States, and the Celsius scale, more widely used in other parts of the world. Here's how it works:

• The constant \(32\) accounts for the freezing point differences between the two scales (0°C = 32°F).
• The fraction \(\frac{9}{5}\) scales the temperature difference between the two scales since a 1°C change equals a 1.8°F change.

To convert a Fahrenheit temperature to Celsius, you need to rearrange the equation to solve for \(C\): \[ C = \frac{5}{9}(F - 32) \]. Simply subtract 32 from the Fahrenheit temperature, multiply by 5, then divide by 9. This calculation shows whether temperatures will be above or below freezing, which can be quite different depending on which scale you use.

temperature range calculation

Understanding how to determine temperature range is essential, especially when dealing with varying daily temperatures. After converting each boundary of the temperature from Fahrenheit to Celsius, the range calculation becomes simple. In this exercise, we started with a low of \(-4^{\circ} F\) and a high of \(23^{\circ} F\). After converting:

• \(-4^{\circ} F\) converts to \(-20^{\circ} C\)
• \(23^{\circ} F\) converts to \(-5^{\circ} C\)

To find the temperature range in Celsius, subtract the lower bound temperature from the upper bound: \[ -5 - (-20) = -5 + 20 = 15 \]The temperature ranged 15 degrees over the period. Calculating this range shows the variance in daily temperatures and helps in understanding how significant or minimal temperature changes are within a given timeframe.

solving linear equations

Solving linear equations is a fundamental mathematical skill, especially when converting temperatures or managing variables. In temperature conversion, you deal with a linear equation of the form\[ F = \frac{9}{5}C + 32 \]. Here's a refresher on how to solve such equations:

• Identify the variable (in this case, \(C\), the Celsius temperature).
• Isolate the variable by performing algebraic operations – subtraction, multiplication, and division – on both sides of the equation.
• Maintain equality by doing the same operation to both sides.

With practice, solving these equations becomes instinctive. If you follow these steps while keeping track of units, especially when dealing with conversions, calculations like converting Fahrenheit values to Celsius degrees will become straightforward. This will also enhance your ability to tackle more complex mathematical problems.