Problem 70 - Precalculus Student Solutions Manual 5th

Question

Temperature in Haiti The average annual temperature in Port-au-Prince, Haiti, is approximately \(28.1^{\circ} \mathrm{C} .\) What is the corresponding Fahrenheit temperature to the nearest tenth of a degree? (Source: www.climatetemp.info/haiti)

Step-by-Step Solution

Answer

The corresponding Fahrenheit temperature is approximately 82.6°F.

Step 1: Understand the Conversion Formula

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

Step 2: Plug in the Given Temperature

Substitute the given Celsius temperature of \(28.1^{\circ} \text{C}\) into the formula: \(F = \frac{9}{5} \times 28.1 + 32\).

Step 3: Perform the Multiplication

Calculate the multiplication part of the formula: \(\frac{9}{5} \times 28.1 = 50.58\).

Step 4: Add 32 to the Result

Add 32 to the result obtained from the multiplication: \(50.58 + 32 = 82.58\).

Step 5: Round to the Nearest Tenth

Round the result to the nearest tenth of a degree: \(82.58\) rounds to \(82.6\).

Key Concepts

Celsius to Fahrenheit

When you need to convert a temperature from Celsius to Fahrenheit, there's a simple formula you can use. The formula is given by: \[ F = \frac{9}{5}C + 32 \]In this formula, \(F\) stands for Fahrenheit and \(C\) stands for Celsius. The goal is to turn the Celsius temperature into Fahrenheit, so you just need to plug in the Celsius value into the formula and do the math. For example, if you have a temperature of \(28.1^{\circ} \, \text{C}\) as in our exercise, plug that into the formula like this: \[ F = \frac{9}{5} \times 28.1 + 32 \] So, the expression becomes \( F = 50.58 + 32 \). Finally, you'll add these numbers together to get the final Fahrenheit temperature.

Temperature Calculation

To get the precise temperature in Fahrenheit, it's important to follow each math step carefully. First, after plugging in the given Celsius temperature, you need to focus on the multiplication. Here, the fraction part means you will multiply 28.1 by 9 and then divide the result by 5. This part looks like this: \[ \frac{9}{5} \times 28.1 = 50.58 \] After the multiplication step, the next step is to add 32 to the result. Why 32? Because the conversion formula requires you to add 32 to adjust for the different starting points of the Celsius and Fahrenheit scales. So now, you add 32 to the multiplication result: \[ 50.58 + 32 = 82.58 \] This final number is the temperature in Fahrenheit, but it needs one more step to be complete.

Rounding Decimals

The last step is to round the Fahrenheit temperature to the nearest tenth. When converting temperatures, it's common to round the final result to a simpler form for easier understanding. In our example, the number 82.58 needs to be rounded to the nearest tenth. To do this, look at the number in the hundredth place - this is right after the tenth place. If this number is 5 or greater (as it is in 82.58), you round up. If it's less than 5, you round down. So, 82.58 rounded to the nearest tenth is 82.6. This means our final converted temperature from 28.1 degrees Celsius is precisely 82.6 degrees Fahrenheit.

• Multiplying by 9/5 and adding 32 adjusts from the metric to imperial scale.
• Rounding helps simplify your final answers.

Keep these steps in mind, and you can convert any Celsius temperature to Fahrenheit accurately.