Problem 58 - Calculus Early Transcendentals: Pearson New International Edition

Question

Fahrenheit temperatures and Celsius temperatures are related by the formula \(C=\frac{5}{9}(F-32)\). An experiment requires that a solution be kept at \(50^{\circ} \mathrm{C}\) with an error of at most \(3 \%\) (or \(1.5^{\circ}\) ). You have only a Fahrenheit thermometer. What error are you allowed on it?

Step-by-Step Solution

Answer

The allowed error on the Fahrenheit thermometer is \(\pm2.7^{\circ}F\).

Step 1: Understanding the error bounds in Celsius

We need to maintain the temperature at \(50^{\circ}C\) with an error of at most \(3\%\). This corresponds to an error of \(1.5^{\circ}C\) as \(3\%\) of \(50\) is \(1.5\). Thus, the temperature in Celsius should be within \(48.5^{\circ}C\) and \(51.5^{\circ}C\).

Step 2: Calculate corresponding Fahrenheit bounds

We use the formula \(C=\frac{5}{9}(F-32)\) to convert the Celsius bounds to Fahrenheit bounds. First, we solve for \(F\) in terms of \(C\): \(F=\frac{9}{5}C+32\). For \(C=48.5\), \(F=\frac{9}{5}(48.5) + 32 \approx 119.3^{\circ}F\). For \(C=51.5\), \(F=\frac{9}{5}(51.5) + 32 \approx 124.7^{\circ}F\). So, the Fahrenheit range is \(119.3^{\circ}F\) to \(124.7^{\circ}F\).

Step 3: Determining the allowable error in Fahrenheit

The target temperature in Celsius is \(50^{\circ}C\), which converts to Fahrenheit as: \(F=\frac{9}{5}(50) + 32 = 122^{\circ}F\). The range we've calculated, \(119.3^{\circ}F\) to \(124.7^{\circ}F\), means the temperature can vary by \(122 - 119.3 = 2.7^{\circ}F\) below and \(124.7 - 122 = 2.7^{\circ}F\) above. This gives an allowable error of \(\pm2.7^{\circ}F\).

Key Concepts

Fahrenheit to Celsius

To effectively convert temperatures from Fahrenheit to Celsius, you’ll use the equation: \[C = \frac{5}{9} \times (F - 32)\].Understanding this formula is key in many scientific and practical applications. It helps translate a Fahrenheit measurement into the Celsius scale, providing an accurate understanding of temperature in different measurement systems.
Here's a breakdown of what each part of the formula does:

• Subtracting 32 adjusts for the offset between the starting points of the Fahrenheit and Celsius scales.
• Multiplying by \(\frac{5}{9}\) converts the measurement across the different relative scales of the two systems.

To reverse the conversion from Celsius to Fahrenheit, you can use the alternate formula: \[F = \frac{9}{5} \times C + 32\]. This is useful when you need to find the temperature in Fahrenheit given a Celsius value. These formulas are incredibly handy when measuring temperatures with different thermometers, especially if you need precise readings for experiments.

Error Analysis

Error analysis is essential when taking any measurement, particularly in a scientific experiment. Errors can occur due to several factors, including instrument limitations, environmental influences, or procedural inaccuracies.
In the context of our temperature conversion exercise, error analysis is specifically concerned with the precision of the temperature measurements. The experiment here stipulates that the solution temperature must be strictly maintained around \(50\,^{\circ}C\) with a maximum allowable deviation of \(3\%\) (or \(1.5\, ^{\circ}C\)). Converting this acceptable error range to Fahrenheit using the aforementioned formulas helps ensure that the thermometer reading remains accurate within the specified limits.
By calculating the allowable error in both scales, you can confidently use a thermometer calibrated in Fahrenheit while still maintaining stringent Celsius temperature requirements. This requires not only conversion precision but also an understanding of the thermometer’s possible errors and ensuring they remain within acceptable bounds.

Thermometer Calibration

Calibration of a thermometer is a crucial process to ensure temperature measurements are accurate and reliable. When dealing with temperature conversions, particularly in scientific experiments, having a well-calibrated thermometer is especially important.
Calibration involves setting a thermometer to known reference points to make sure it reads accurately across its intended range. For example, a thermometer might be calibrated at the freezing and boiling points of water.
Here are some steps involved in calibration:

• Identify fixed points like ice-water mixture (\(0^{\circ}C\)) and boiling water (\(100^{\circ}C\)).
• Compare the readings of the thermometer with these fixed points and adjust as necessary.
• Periodically recalibrate your thermometer to prevent drift over time.

When using a Fahrenheit thermometer in experiments requiring Celsius accuracy, calibration ensures the temperature is consistently monitored within the accepted error margin (\(\pm2.7^{\circ}F\) in this example). Moreover, regular calibration checks, especially before sensitive measurements, can prevent costly errors.