Question

In the following exercises, substitute the given quantitics into the indicated formula from technology and finance. Use the formula \(C=\frac{5}{9}(F-32)\) to convert \(128^{\circ} \mathrm{F}\) to degrees Celsius.

Short answer

\(128^\circ F\) is approximately \(53.33^\circ C\).

Step-by-step solution

Identify the Formula for Conversion

Recognize the given temperature conversion formula from Fahrenheit to Celsius, which is \(C = \frac{5}{9}(F - 32)\), where \(C\) represents degrees Celsius and \(F\) represents degrees Fahrenheit.

Substitute the Given Fahrenheit Temperature

Substitute the given Fahrenheit temperature, \(128^\circ F\), into the conversion formula to find the equivalent temperature in Celsius. The substitution results in \(C = \frac{5}{9}(128 - 32)\).

Perform the Calculation

Subtract 32 from 128 to find the temperature in degrees above the freezing point of water in the Fahrenheit scale. This gives us \(C=\frac{5}{9}(96)\). Then, multiply 96 by \(\frac{5}{9}\) to obtain the Celsius temperature. After performing the multiplication, that results in \(C=53.333...^\circ C\).

Round off the Result (if necessary)

Based on the context or instructions given for rounding, round the calculated Celsius temperature to the desired number of decimal places. Here, if we round to two decimal places, we get \(C=53.33^\circ C\).

Key concepts

Fahrenheit to Celsius Formula

The magic behind turning Fahrenheit degrees into Celsius lies in the simple yet elegant formula: \(C = \frac{5}{9}(F - 32)\). To understand how this formula came to be, one must know that the Fahrenheit and Celsius scales fix the freezing and boiling points of water at different values. In Celsius, water freezes at 0°C and boils at 100°C, creating a 100-degree interval. Fahrenheit, on the other hand, sets these points at 32°F and 212°F, making a 180-degree interval between the two. The ratio of 5/9 is derived from this relationship; 100 degrees Celsius equal 180 degrees Fahrenheit. Subtracting the 32 offsets the fact that the scales start at different values for the freezing point of water.

When using this formula, remember that the order of operations matter. Always subtract 32 first, since this shifts the scale, and then apply the ratio to convert to Celsius degrees. It’s not simply a mathematical exercise but a practical way to bridge two predefined scales of temperature measurement.

Unit Conversion

Unit conversion is a fundamental skill not just in temperature measurement but across various fields including science, engineering, and daily life. It involves changing the measurement of a quantity from one unit to another without changing the actual amount. Converting temperature from Fahrenheit to Celsius is a common example, but the concept applies to length (miles to kilometers), volume (gallons to liters), or even currency (dollars to euros).

The key to successful unit conversion is understanding the relationship between different units and applying the appropriate conversion factor. The conversion factor is a known quantity used to convert from one unit to another through multiplication or division. It's essential to ensure that the units being converted are of the same dimension or measure the same kind of physical quantity to avoid erroneous results.

Applied Mathematics

Applied mathematics infiltrates our lives, often without our conscious realization, solving real-world problems through mathematical techniques. Temperature conversion is an excellent example of applied mathematics at work. It encompasses the use of algebraic formulas to make meaningful translations between different units of measurement.

In temperature conversion, applied mathematics allows us to comprehend and navigate the differences between temperature scales used in various regions or scientific contexts. By demystifying the formula \(C = \frac{5}{9}(F - 32)\) and applying it to real scenarios, such as cooking or weather forecasting, we utilize applied mathematics to adapt information for practical use. Understanding the underlying mathematics gives us the power to perform a variety of tasks, ranging from simple daily conversions to complex scientific computations with confidence and precision.