Problem 77 - Algebra and Trigonometry with Analytic Geometry

Question

Temperature readings on the Fahrenheit and Celsius scales are related by the formula \(C=\frac{5}{9}(F-32)\). What values of \(F\) correspond to the values of \(C\) such that \(30 \leq C \leq 40\) ?

Step-by-Step Solution

Answer

The values of \(F\) are from 86 to 104 Fahrenheit.

Step 1: Identify the Relationship

We are given the formula that relates Celsius (\(C\)) and Fahrenheit (\(F\)): \(C = \frac{5}{9}(F - 32)\). We need to determine \(F\) for \(C\) values between 30 and 40.

Step 2: Solve for Integer Endpoints

First, solve for \(F\) when \(C = 30\): \[30 = \frac{5}{9}(F - 32)\]. Multiplying both sides by 9 to clear the fraction, we get: \[270 = 5(F - 32)\]. Divide by 5 to obtain: \[54 = F - 32\]. Adding 32 to both sides gives us: \[F = 86\].

Step 3: Solve for Upper Endpoint

Now solve for \(F\) when \(C = 40\): \[40 = \frac{5}{9}(F - 32)\]. Multiply both sides by 9 to clear the fraction, resulting in: \[360 = 5(F - 32)\]. Divide by 5: \[72 = F - 32\]. Adding 32 to both sides, we have: \[F = 104\].

Step 4: Identify Range of F

We have found that when \(C = 30\), \(F = 86\), and when \(C = 40\), \(F = 104\). Therefore, the range of \(F\) values corresponding to \(30 \leq C \leq 40\) is from 86 to 104.

Key Concepts

Temperature Conversion Formula

The process of converting temperatures from Fahrenheit to Celsius and vice versa is crucial in many scientific and everyday applications. The formula to convert Fahrenheit (\(F\)) to Celsius (\(C\)) is:
\[C = \frac{5}{9}(F - 32)\]Here’s how it works:

• Subtract 32 from the Fahrenheit temperature. This adjusts for the Fahrenheit scale's zero point.
• Multiply the result by \(\frac{5}{9}\). This factor comes from the ratio of degree sizes between the Celsius and Fahrenheit scales.

This conversion formula derives from the linear relationship where each degree Celsius changes by \(\frac{5}{9}\) of a degree Fahrenheit. This formula is widely used, especially in regions transitioning between the two systems of measurement.

Linear Equations

A linear equation is a type of equation where each term is a constant or the product of a constant and a single variable. In the temperature conversion problem, we use the linear equation:
\[C = \frac{5}{9}(F - 32)\]This equation can be rearranged to solve for another variable. When solving, we follow a series of steps:

• First, isolate the variable of interest. In our case, if solving for \(F\), rearrange the equation accordingly.
• Use inverse operations to simplify. If there’s a fraction, multiply by its reciprocal to clear it.
• Perform similar steps on either side of the equation to maintain equality.

Once you grasp solving linear equations, you can easily manipulate and find values for different scenarios, just like finding Celsius temperatures that fit specific Fahrenheit ranges.

Inequalities

Inequalities help us understand a range of possible values for variables rather than a single solution. For the exercise at hand, we are dealing with Celsius temperature values ranging between 30 and 40:
\[30 \leq C \leq 40\]Here’s how to handle inequalities:

• Conduct operations on each side exactly as you would in an equation, ensuring to flip the inequality sign when multiplying or dividing by a negative number.
• Treat the ends individually - solve separately for lower and upper bounds.
• Combine solutions to determine the overall range for the variable you seek.

By solving these inequalities step by step, we linearly map out the Fahrenheit equivalents that correspond to the Celsius range, using our conversion formula.