Question

Convert temperatures as indicated. Round your answer to the nearest tenth. \(39.8^{\circ} \mathrm{C}=\) _______ \({ }^{\circ} \mathrm{F}\)

Short answer

\(39.8^{\circ} \mathrm{C} = 103.6^{\circ} \mathrm{F}\).

Step-by-step solution

Understand the formula

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

Insert the known value

Insert the Celsius temperature given in the problem into the formula. In this case, it is \( 39.8^{\circ} \mathrm{C} \). The formula becomes: \[ F = \frac{9}{5} \times 39.8 + 32 \]

Perform the multiplication

First, multiply \( 39.8 \) by \( \frac{9}{5} \) to convert the Celsius temperature. Calculate: \[ \frac{9}{5} \times 39.8 = 71.64 \]

Add 32

Add 32 to the result from the previous calculation: \[ 71.64 + 32 = 103.64 \] This is the temperature in Fahrenheit before rounding.

Round the answer

Round the temperature to the nearest tenth as instructed. \( 103.64 \) rounds to \( 103.6 \) when rounded to the nearest tenth.

Key concepts

Celsius to Fahrenheit

Converting Celsius temperatures to Fahrenheit is a common procedure, crucial for activities ranging from baking to scientific experimentation. It involves a mathematical formula that acts as a bridge between these two scales. To understand the conversion, you first need to appreciate the differences between Celsius and Fahrenheit.

• Celsius scale is based on the freezing (0°C) and boiling points (100°C) of water.
• Fahrenheit scale instead, fixes water freezing at 32°F and boiling at 212°F.

The conversion formula, \( F = \frac{9}{5}C + 32 \), allows you to calculate the Fahrenheit equivalent of a given Celsius temperature. By multiplying the Celsius temperature by \( \frac{9}{5} \), and then adding 32, you effectively scale and shift Celsius values to fit the Fahrenheit range.

Temperature Formulas

Temperature conversion requires a reliable formula, which in this case maps Celsius values into the Fahrenheit spectrum. The formula used, \( F = \frac{9}{5}C + 32 \), is grounded on the proportional relationship between the scales.

• The \( \frac{9}{5} \) factor converts the Celsius degree increment to Fahrenheit’s.
• Adding 32 adjusts for the difference in the starting point of each scale (freezing point of water).

This formula is crucial not just for basic conversions, but also plays a significant role in scientific calculations and in regions where both temperature scales are in use. Understanding how and why this formula works deepens the comprehension of linear relationships in math, aligning temperature conversion with everyday practical uses.

Mathematics Problem-Solving

Solving temperature conversion problems reflects the wider process of mathematical problem-solving, which involves interpreting and applying formulas. It demands a systematic approach:

• Start by understanding the formula and identifying known and unknown variables.
• Substitute the known values into the algebraic formula.
• Perform arithmetic operations step by step, caring for accuracy.
• Finally, interpret the numeric solution and apply any necessary rounding.

In our example, beginning with the known Celsius temperature, you apply each step of the formula: multiplying, then adding, and lastly rounding. This systematic approach doesn't just help in conversions but is a cornerstone of problem-solving in mathematics. Grasping these steps ensures students can tackle similar tasks independently, fostering both mathematical understanding and analytical skill development.