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.) A nurse reports a temperature of \(37.8^{\circ} \mathrm{C}\). _______ \({ }^{\circ} \mathrm{F}\)

Short answer

The equivalent temperature is 100.0°F.

Step-by-step solution

Understanding Conversion Formula

To convert a temperature from degrees Celsius (\(^{\circ} \mathrm{C}\)) to degrees Fahrenheit (\(^{\circ} \mathrm{F}\)), we use the formula: \[ T_{\mathrm{F}} = \frac{9}{5}T_{\mathrm{C}} + 32 \]where \(T_{\mathrm{C}}\) is the temperature in Celsius, and \(T_{\mathrm{F}}\) is the temperature in Fahrenheit.

Substitute the Given Temperature

Substitute \(37.8^{\circ} \mathrm{C}\) for \(T_{\mathrm{C}}\) in the conversion formula: \[ T_{\mathrm{F}} = \frac{9}{5} \times 37.8 + 32 \]

Calculate the Conversion

First, calculate the product \(\frac{9}{5} \times 37.8\):\[ \frac{9}{5} \times 37.8 = 68.04 \]Next, add 32 to the result:\[ 68.04 + 32 = 100.04 \]

Round the Result

Round the result to the nearest tenth of a degree, if necessary. Here, 100.04 rounds to:\[ 100.0 \]

Key concepts

Celsius to Fahrenheit

When we talk about the Celsius to Fahrenheit conversion, we are moving between two different scales of temperature measurement. The Celsius scale is commonly used worldwide, except in the United States, where Fahrenheit is more common. Understanding this conversion is crucial, especially for fields like science and healthcare, where precise communication about temperature is important.

The formula used for this conversion helps you calculate the equivalent temperature in Fahrenheit when you know the temperature in Celsius. It’s a simple, linear equation given by:

• \[ T_{\mathrm{F}} = \frac{9}{5}T_{\mathrm{C}} + 32 \]

Here, \(T_{\mathrm{C}}\) stands for the Celsius temperature, while \(T_{\mathrm{F}}\) is what we’re calculating — the temperature in Fahrenheit. This formula derives from the fact that the freezing point of water is 0°C, which is 32°F, and the boiling point is 100°C, which equals 212°F, thus maintaining the proportional relationships. After understanding how these two points define our linear conversion, performing the actual calculation becomes straightforward.

mathematical formula

Mathematical formulas are tools that allow us to work with numbers in a consistent way. In the case of converting Celsius to Fahrenheit, the formula \[ T_{\mathrm{F}} = \frac{9}{5}T_{\mathrm{C}} + 32 \] acts as a guideline for this conversion.

Let's break down the function step-by-step:

• Start by multiplying the given temperature in Celsius by \(\frac{9}{5}\). This converts the Celsius value to the equivalent on the Fahrenheit scale, disregarding offset.
• Afterward, you add 32. This accounting for the fact that 0°C aligns with 32°F, thereby shifting the scale to match practical temperature readings used in everyday life.

Even though this formula is straightforward, practicing it can cement your understanding every time you need a temperature conversion. Plugging values into this equation should become second nature as you work through similar problems and see how consistently the steps apply.

rounding numbers

Rounding numbers is a mathematical process that simplifies a number to make it easier to work with, especially when only a certain level of precision is needed. In the exercise example, we calculated the temperature in Fahrenheit to be 100.04°F.

When asked to round a number to the nearest tenth, you follow these steps:

• Look at the hundredths place in the decimal. Here, in 100.04, the digit is 4.
• If the digit is 4 or less, you simply drop it and leave the tenths digit as is. In this case, the rounded number becomes 100.0.
• If it were 5 or more, you would increase the tenths digit by one and drop the remaining digits.

This rounding helps us communicate more effectively by focusing on meaningful precision. It's an essential skill in everyday math, science, and any area where numerical precision matters.