Problem 67 - Algebra A Combined Function

Question

The average temperature on the planet Mercury is \(167^{\circ} \mathrm{C}\). Convert this temperature to degrees Fahrenheit. Round to the nearest degree.

Step-by-Step Solution

Answer

The temperature on Mercury is approximately 333°F.

Step 1: Understand the Conversion Formula

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

Step 2: Substitute the Celsius Temperature into the Formula

Substitute \(C = 167^{\circ} \mathrm{C}\) into the given conversion formula:\[F = \left( \frac{9}{5} \right)(167) + 32\]

Step 3: Perform the Multiplication

Calculate the product of \(\frac{9}{5}\) and \(167\):\[F = \left( \frac{9}{5} \right)(167) = 300.6\]

Step 4: Add 32

Now add 32 to the previous result to complete the conversion:\[F = 300.6 + 32 = 332.6\]

Step 5: Round to the Nearest Degree

Finally, round 332.6 to the nearest whole number to get the temperature in degrees Fahrenheit:\[F \approx 333^{\circ} \mathrm{F}\]

Key Concepts

Celsius to Fahrenheit

Converting temperatures from Celsius to Fahrenheit is useful for understanding temperature in different contexts, especially when using different measurement systems. The Celsius to Fahrenheit conversion can be easily accomplished with a simple mathematical relationship.

The formula to convert Celsius (\(C\)) to Fahrenheit (\(F\)) is:

• \[ F = \left( \frac{9}{5} \right)C + 32 \]

This formula helps us convert temperatures expressed in degrees Celsius to the Fahrenheit scale, which is often used in countries like the United States.
The factor \(\frac{9}{5}\) is used because there are 9 Fahrenheit degrees for every 5 Celsius degrees. The value 32 is added because \(0^{\circ}\) Celsius is equivalent to \(32^{\circ}\) Fahrenheit, the freezing point of water.

Using this formula, you input Celsius temperature values to find their Fahrenheit equivalents. In our example, Mercury's average temperature \(167^{\circ}\) Celsius becomes a different value when converted to Fahrenheit.

mathematical formula

Mathematical formulas serve as a guide for performing calculations and solving problems efficiently. The specific formula for converting Celsius to Fahrenheit mentioned here connects the Celsius temperature scale to the Fahrenheit scale.

When using any mathematical formula, understanding each component and its purpose is vital:

• The fraction \(\frac{9}{5}\) establishes the rate of conversion between two scales, accounting for the difference in scale units.
• Every formula element, like the addition of 32 in this case, aligns the two temperature systems exactly at the freezing point of water.

Using formulas involves substitution and arithmetic calculations. For instance, in our example, inserting 167 for \(C\) involves:

• Multiplying 167 by \(\frac{9}{5}\), yielding 300.6.
• Adding 32 to the result to achieve the proper Fahrenheit temperature.

This structured approach helps ensure accuracy and understanding when shifting between measurement systems. The usability of a formula like this is significant for students learning about scientific measurements and data handling.

rounding numbers

Rounding numbers is a frequent task in mathematics, helping make numbers easier to work with or communicate. It's an essential skill for scenarios where precision can be simplified, like temperature conversion.

In this exercise, rounding was necessary to express the calculated Fahrenheit temperature in a more practical, whole number form:

• After converting \(167^{\circ}\) Celsius to \(332.6^{\circ}\) Fahrenheit, rounding became the final step.

Rounding to the nearest degree means looking at the first decimal place. If it's 5 or more, we round up; if less than 5, we round down. Hence the \(332.6\) became \(333^{\circ}\).

This step simplifies data presentation and avoids unnecessary detail in practical use. For scientists, engineers, and students alike, mastering rounding is part of dealing with real-world measurements where exact precision isn't usually needed. Understanding when and how to round numbers is a key aspect of working with data efficiently.