Problem 95 - College Algebra

Question

The formula for converting Fahrenheit temperature, \(F\) to Celsius temperature, \(C\), is $$C=\frac{5}{9}(F-32)$$ If Celsius temperature ranges from \(15^{\circ}\) to \(35^{\circ},\) inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.

Step-by-Step Solution

Answer

When converted, the Celsius range of \(15^{\circ}\) to \(35^{\circ}\) equates to a Fahrenheit range of \([59^{\circ}, 95^{\circ}]\).

Step 1: Convert Lower Bound of Celsius Range to Fahrenheit

Apply the temperature conversion formula to the lower bound of the Celsius range, which is \(15^{\circ}\). The conversion formula is \(C = \frac{5}{9}(F - 32)\), but to find the Fahrenheit temperature from a known Celsius temperature, it must be rearranged: \(F = \frac{9C}{5} + 32 \). Substituting \(15^{\circ}\) for \(C\), we compute \(F = \frac{9(15)}{5} + 32\).

Step 2: Convert Upper Bound of Celsius Range to Fahrenheit

Next, apply the temperature conversion formula in the rearranged form to the upper bound of the Celsius range, which is \(35^{\circ}\). Substituting \(35^{\circ}\) for \(C\), compute \(F = \frac{9(35)}{5} + 32\).

Step 3: Express Fahrenheit Range in Interval Notation

Now that both the lower and upper bounds of the Fahrenheit range have been calculated, express the range using interval notation, which is shown as [lower bound, upper bound].

Key Concepts

Temperature Conversion Formula

Converting temperatures between Celsius and Fahrenheit is a common mathematical task, especially in science and weather forecasting. The temperature conversion formula helps translate temperatures from one system to another. The primary formula for converting Fahrenheit (F) to Celsius (C) is: \( C = \frac{5}{9}(F - 32) \) This formula subtracts 32 from the Fahrenheit temperature, because 0°C corresponds to 32°F on the two scales. Then, it multiplies the result by the fraction \( \frac{5}{9} \) since each Celsius degree equals \( \frac{5}{9} \) of a Fahrenheit degree. To convert a Celsius temperature back to Fahrenheit, we use a rearranged version of the formula: \( F = \frac{9C}{5} + 32 \) Here, we multiply the Celsius temperature by \( \frac{9}{5} \), since each Fahrenheit degree equals \( \frac{9}{5} \) of a Celsius degree, then add 32 to adjust from the Celsius offset.

Fahrenheit Temperature Calculation

Calculating the Fahrenheit temperature from a given Celsius temperature involves using the rearranged conversion formula mentioned earlier. To obtain the Fahrenheit degree, substitute the known Celsius temperature into: \( F = \frac{9C}{5} + 32 \) For example, to find the Fahrenheit equivalent of 15°C, substitute 15 for C: - Calculate \( \frac{9 \times 15}{5} \) which is equal to 27. - Add 32 to the result. This gives us 59°F. Similarly, to convert 35°C: - Calculate \( \frac{9 \times 35}{5} \), resulting in 63. - Add 32 to get 95°F. Thus, 15°C converts to 59°F and 35°C transforms to 95°F. These calculations show how precise and straightforward using the formula can be.

Interval Notation in Mathematics

Interval notation is a mathematical method used to represent a range of values between two endpoints. It is particularly useful in expressing temperature ranges, as it is concise and clear. This notation uses brackets and parentheses to show inclusivity or exclusivity of the endpoints: - Square brackets \([ ]\) indicate that the endpoint is included (known as inclusive). - Parentheses \(( )\) indicate that the endpoint is not included (known as exclusive). In the example problem, we calculated the Fahrenheit equivalents for the Celsius interval from 15°C to 35°C. With results of 59°F and 95°F, we use inclusive brackets since both endpoints are part of the range. The interval is expressed as: - \([59, 95]\) This notation efficiently captures the entire range of temperatures in one compact expression.