Question

For each of the following statements, change the given temperature to its corresponding equivalent in \({ }^{\circ} \mathrm{C}\) or \({ }^{\circ} \mathrm{F}\). (Round to the nearest tenth.) Do not expose a medication to temperatures greater than \(84^{\circ} \mathrm{F}\). _______ \({ }^{\circ} \mathrm{C}\)

Short answer

84°F is approximately 28.9°C.

Step-by-step solution

Identify the conversion formula

To convert temperatures from degrees Fahrenheit (°F) to degrees Celsius (°C), we use the formula: \[ C = \frac{5}{9}(F - 32) \] where \( C \) is the temperature in Celsius and \( F \) is the temperature in Fahrenheit.

Substitute the given temperature into the formula

Let's substitute the temperature given in the problem, which is \( F = 84° \), into the formula. This gives us: \[ C = \frac{5}{9}(84 - 32) \]

Calculate the result

First, calculate the difference inside the parentheses: \( 84 - 32 = 52 \). Now, substitute back into the formula: \[ C = \frac{5}{9} \times 52 \] Next, multiply: \( 5 \times 52 = 260 \). Then, divide by 9: \( \frac{260}{9} \approx 28.888... \).

Round the result to the nearest tenth

We need to round 28.888... to the nearest tenth. The digit in the tenths place is 8, and the subsequent digit is also 8, which means we round up. Thus, \( C \approx 28.9^{\circ} \mathrm{C} \).

Key concepts

Mathematical Formulas

In mathematics, formulas serve as powerful tools to solve problems consistently and accurately. A formula is an equation that shows the relationship between different variables.
It acts like a blueprint to guide us on how to perform calculations.

Temperature conversion is a prime example where mathematical formulas are utilized frequently. The formula to convert Fahrenheit to Celsius is:

• \( C = \frac{5}{9}(F - 32) \)

This formula relates the Fahrenheit temperature \( F \) to its Celsius equivalent \( C \). By substituting known values into equations like this, we can easily compute the desired result.
This process not only saves time but also ensures precision in our calculations.
Understanding the origin and application of such formulas enhances confidence in solving various mathematical tasks.

Fahrenheit to Celsius

Switching from Fahrenheit to Celsius is a fundamental exercise in temperature conversion. Many regions in the world use the Celsius scale, making it essential to grasp this conversion.
Understanding how this works goes beyond switching scales—it involves recognizing their distinct starting points and intervals.

In the Fahrenheit scale, water freezes at 32 degrees and boils at 212 degrees.
In contrast, with the Celsius scale, water freezes at 0 degrees and boils at 100 degrees.
This significant difference requires a conversion formula to bridge the levels accurately. To transform Fahrenheit into Celsius, remember this essential formula:

• \( C = \frac{5}{9}(F - 32) \)

This formula subtracts 32 from the Fahrenheit value to adjust for the freezing point difference and then multiplies by \( \frac{5}{9} \) to account for the scale disparity.
Practicing these conversions regularly will help reinforce the method so it becomes second nature.

Temperature Rounding

Rounding is a mathematical technique used to simplify numbers, making them easier to understand or use in further calculations. It becomes essential in real-world applications like temperature mapping where precision needs to be balanced with clarity.
When dealing with temperatures, rounding allows us to communicate conditions without unnecessary detail.

In our example, after calculating the Celsius equivalent of 84°F, we arrived at a number like 28.888....
Since dealing with too many decimal places can be cumbersome, it's generally useful to round this number.
The concept of rounding to the nearest tenth focuses on the tenths decimal place.
Here's the basic rule:

• If the digit right after the tenths is 5 or more, round up the tenths.
• If it’s less than 5, leave the tenths as is.

For 28.888..., since the first decimal after 8 is 8 itself, we round up to get 28.9°C.
This simpler representation helps communicate the information succinctly while retaining practical accuracy.