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?

Short answer

The allowed Fahrenheit error is \(\pm 2.7^{\circ}F\).

Step-by-step solution

Convert Celsius to Fahrenheit

Use the given formula to convert the target temperature from Celsius to Fahrenheit. Start by setting \(C = 50^{\circ}C\) and solve for \(F\) using the formula \(C=\frac{5}{9}(F-32)\). Rearranging gives \(F = \frac{9}{5}C + 32\). Substitute \(C = 50\) to find \[ F = \frac{9}{5} \times 50 + 32 = 122^{\circ}F. \] This is the target Fahrenheit temperature.

Calculate Permissible Celsius Error

According to the problem, the solution must be within \(1.5^{\circ}C\) of the target \(50^{\circ}C\). This defines the permissible Celsius error: \(\pm 1.5^{\circ}C\).

Convert Celsius Error to Fahrenheit Error

Determine the corresponding Fahrenheit errors for the given Celsius error range. Use \(C = \frac{5}{9}(F-32)\) with \(C = 51.5^{\circ}C\) and \(C = 48.5^{\circ}C\) to find the Fahrenheit equivalents: 1. For \(51.5^{\circ}C\): \[ F = \frac{9}{5} \times 51.5 + 32 = 124.7^{\circ}F. \] 2. For \(48.5^{\circ}C\): \[ F = \frac{9}{5} \times 48.5 + 32 = 119.3^{\circ}F. \] Thus, the permissible Fahrenheit range is from \(119.3^{\circ}F\) to \(124.7^{\circ}F\).

Determine Allowed Fahrenheit Error

Subtract the calculated Fahrenheit extremes from the target Fahrenheit temperature \(122^{\circ}F\). The maximum error allowed is \(122^{\circ}F - 119.3^{\circ}F = 2.7^{\circ}F\). Thus, the thermometer should not have an error exceeding \(\pm 2.7^{\circ}F\).

Key concepts

Fahrenheit to Celsius

Understanding the conversion from Fahrenheit to Celsius is essential when dealing with temperature measurements in different units. This conversion lets you adapt temperatures from the Fahrenheit scale, often used in the United States, to the Celsius scale, which is prevalent in most of the world.

The formula for converting a Fahrenheit temperature to Celsius is:

• \(C = \frac{5}{9}(F - 32)\)

This equation works by first subtracting 32, the offset between the freezing points of water in the two scales, and then adjusting for the differing size of their degrees. Each degree of Fahrenheit equals \(\frac{5}{9}\) of a Celsius degree, making this formula crucial for accurate conversions.

If you have a Fahrenheit temperature, simply plug it into the formula to find the equivalent Celsius degree. For instance, if the Fahrenheit thermometer reads \(122^{\circ}F\), the equivalent Celsius temperature is calculated as:

• \(C = \frac{5}{9}(122 - 32) = 50^{\circ}C\)

Celsius to Fahrenheit

To convert a temperature from Celsius to Fahrenheit, we use a rearranged version of the previous formula. This conversion is useful when working with equipment or materials specified in Fahrenheit.

The conversion equation is:

• \(F = \frac{9}{5}C + 32\)

In this formula, \(\frac{9}{5}\) accounts for the larger size of a degree on the Fahrenheit scale, and adding 32 shifts the temperature to match the offset between the two scales' freezing points.

If you need to find the Fahrenheit equivalent of a Celsius temperature, for example, \(50^{\circ}C\), use the formula:

• \(F = \frac{9}{5} \times 50 + 32 = 122^{\circ}F\)

This shows how you can quickly convert any given Celsius temperature into Fahrenheit using this simple method.

Error Analysis in Temperature Measurement

Error analysis is critical in ensuring accuracy in temperature measurements, especially in scientific experiments requiring precise conditions.

For example, if the target temperature is \(50^{\circ}C\) with a permissible error of \(\pm 1.5^{\circ}C\), the allowable Celsius temperature range is from \(48.5^{\circ}C\) to \(51.5^{\circ}C\). To find the equivalent Fahrenheit error, both endpoints of the Celsius range are converted to Fahrenheit.

Converting \(51.5^{\circ}C\) results in \(124.7^{\circ}F\), while \(48.5^{\circ}C\) gives \(119.3^{\circ}F\). Therefore, the Fahrenheit temperature should range from \(119.3^{\circ}F\) to \(124.7^{\circ}F\), providing an allowable error of \(\pm 2.7^{\circ}F\) around the \(122^{\circ}F\) target.

Such precision in defining the margin of error helps prevent significant discrepancies that could affect experiment outcomes. It's vital to recognize this range for the integrity of data and results.

Formula for Temperature Conversion

Temperature conversion formulas bridge the gap between different measurement systems. They allow seamless translation of temperature values between Celsius and Fahrenheit scales.

The two primary formulas are:

• From Fahrenheit to Celsius: \(C = \frac{5}{9}(F - 32)\)
• From Celsius to Fahrenheit: \(F = \frac{9}{5}C + 32\)

Using these formulas, you can easily convert any temperature from one system to the other, essential for international communication and documentation.

It's important to understand these formulas thoroughly as they ensure that measurements remain consistent and comparable, no matter the origin of the data. Whether you're working in a laboratory, cooking, or tracking the weather, this knowledge facilitates exact measurement conversions.