Problem 124 - Introductory and Intermediate Algebra for College Students 4th

Question

The formula $$C=\frac{5}{9}(F-32)$$ expresses the relationship between Fahrenheit temperature, \(F\) and Celsius temperature, \(C .\) In Exercises \(123-124,\) use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale. $$41^{\circ} \mathrm{F}$$

Step-by-Step Solution

Answer

The temperature \( 41^\circ F \) is equivalent to \( 5^\circ C \)

Step 1: Substitute the given Fahrenheit temperature in the formula

Substitute \( F = 41 \) degrees Fahrenheit into the Celsius conversion formula to set up the equation to find \(C\), the equivalent temperature in Celsius. This yields \( C = \frac{5}{9}(41-32) \) degrees Celsius.

Step 2: Simplify inside the bracket

Simplify inside the parenthesis, yielding \( C = \frac{5}{9}(9) \) degrees Celsius.

Step 3: Multiply the temperatures

Multiply the fraction by 9 to obtain the equivalent temperature in Celsius. This gives \( C = 5 \) degrees Celsius.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, operators, and variables. They are an essential element in a wide range of mathematical problems, including temperature conversion. An expression, however, is distinct because it does not contain an equality sign, as opposed to equations which do.

For example, in the Fahrenheit to Celsius conversion formula, \( C = \frac{5}{9}(F - 32) \), \( C \) and \( F \) are variables that represent the temperatures in Celsius and Fahrenheit, respectively. The expression \( \frac{5}{9}(F - 32) \) includes the numbers 5 and 9, the variable \( F \) representing the Fahrenheit temperature, and the arithmetic operators \( \frac{\underline{\phantom{xx}}}{\underline{\phantom{xx}}} \) (division) and \( - \) (subtraction). Understanding how to manipulate this algebraic expression is crucial in converting temperatures from Fahrenheit to Celsius effectively.

Temperature Conversion

Temperature conversion is a process through which we translate a temperature value from one unit system to another, such as from Fahrenheit to Celsius or vice versa. It's an important skill, as different countries and fields of science use different temperature scales.

The formula \( C = \frac{5}{9}(F - 32) \) is used to convert a Fahrenheit (\( F \) ) temperature reading to Celsius (\( C \) ). The constants in this formula, 5/9 and 32, are derived from the relationship between the two scales and ensure that the conversion is accurate. The number 32 represents the difference in the freezing point of water on both scales, while the fraction 5/9 accounts for the ratio of the two scales' increments.

It is important to note that an exact Fahrenheit temperature equivalent can sometimes result in a decimal or fractional Celsius temperature, and depending on the context, it may require rounding to a practical number of decimal places.

Substitution Method

The substitution method is an algebraic technique used to solve systems of equations or to simplify expressions by replacing variables with known or assumed values. In the context of our temperature conversion problem, we use the substitution method to find the Celsius temperature equivalent of a given Fahrenheit temperature.

To illustrate, when given a Fahrenheit temperature of 41 degrees, we substitute \( F = 41 \) into the conversion formula. This means we replace \( F \) with 41 in the expression, yielding the new expression \( \frac{5}{9}(41 - 32) \). No more variables exist in the expression, and we can perform arithmetic to simplify it to a single numerical value that represents the Celsius temperature.

The substitution method is not only useful for temperature conversions but is also a fundamental concept in algebra that is applied in various mathematical scenarios, such as solving equations or evaluating expressions.