Problem 69 - Precalculus Student Solutions Manual 5th

Question

Temperature in Montreal The average low temperature in Montreal, Canada, for the date January 30 is \(7^{\circ} \mathrm{F}\). What is the corresponding Celsius temperature to the nearest tenth of a degree? (Source: www.wunderground.com)

Step-by-Step Solution

Answer

-13.9 °C

Step 1 - Understand the formula for temperature conversion

The formula to convert Fahrenheit to Celsius is given by: \[ C = \frac{5}{9} (F - 32) \]where \(C\) is the temperature in Celsius and \(F\) is the temperature in Fahrenheit.

Step 2 - Substitute the given Fahrenheit temperature

Substitute \(F = 7\) into the conversion formula: \[ C = \frac{5}{9} (7 - 32) \]

Step 3 - Perform the subtraction inside the parentheses

Calculate \(7 - 32\): \[ 7 - 32 = -25 \]

Step 4 - Multiply the result by \( \frac{5}{9} \)

Now, convert to Celsius: \[ C = \frac{5}{9} (-25) \approx -13.888 \]

Step 5 - Round to the nearest tenth

Finally, round \(-13.888\) to the nearest tenth: \[ C \approx -13.9 \]

Key Concepts

Fahrenheit to Celsius

One of the fundamental concepts in temperature conversion is understanding how to change temperatures from Fahrenheit to Celsius. The Fahrenheit and Celsius scales are two of the most common temperature scales used around the world. The key difference between them is their starting points and increments.

The Celsius scale is based on water freezing at 0°C and boiling at 100°C. The Fahrenheit scale sets water freezing at 32°F and boiling at 212°F. To convert a temperature from Fahrenheit to Celsius, we use a specific formula. Understanding this formula is crucial for accurate conversions.

temperature conversion formula

The formula to convert Fahrenheit to Celsius is:

\[C = \frac{5}{9} (F - 32)\]

- **C** is the temperature in Celsius.
- **F** is the temperature in Fahrenheit.

Here's what each part of the formula means:
- First, you subtract 32 from the Fahrenheit temperature (\(F - 32\)). This is because the Fahrenheit scale starts at 32°F for the freezing point of water, while the Celsius scale starts at 0°C.
- Next, you multiply the result by \(\frac{5}{9}\) to adjust for the difference in scale increments. The scales are different not only in where they start but also in the size of each degree.

For example, let's convert 7°F to Celsius:
1. Substitute 7 for **F** in the formula: \[C = \frac{5}{9} (7 - 32)\]
2. Subtract 32 from 7: \(7 - 32 = -25\)
3. Multiply the result by \(\frac{5}{9}\): \[\frac{5}{9} (-25) = -13.888\]

Now, we have the temperature in Celsius as -13.888.

rounding decimals

The final step in the process is rounding the result to the nearest tenth. Rounding decimals is an important skill because it helps simplify numbers in a way that's still accurate enough for most practical purposes.

In our example, we calculated \(-13.888\) degrees Celsius.

Rounding rules to remember:
- Look at the number in the hundredths place (second digit after the decimal).
- If this number is 5 or higher, round the number in the tenths place (first digit after the decimal) up by one.
- If this number is 4 or lower, keep the number in the tenths place the same.

For \(-13.888\):
- The hundredths place is 8 (since the number is \(-13.888\)).
- Because 8 is greater than 5, we round the tenths place up from 8 to 9.

Therefore, \(-13.888\) rounded to the nearest tenth is \(-13.9\). This rounded value is more manageable for most uses while remaining accurate.