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Chapter 1: Problem 97

A column of liquid is found to expand linearly on heating. Assume the column rises \(5.25 \mathrm{~cm}\) for a \(10.0^{\circ} \mathrm{F}\) rise in temperature. If the initial temperature of the liquid is \(98.6^{\circ} \mathrm{F}\), what will the final temperature be in \({ }^{\circ} \mathrm{C}\) if the liquid has expanded by \(18.5 \mathrm{~cm} ?\)

Short Answer

Expert verified
The final temperature of the liquid in Celsius is approximately \(38.47^{\circ}\mathrm{C}\).

Step by step solution

01

Calculate the temperature increase in Fahrenheit

First, we need to find out the temperature increase in Fahrenheit. We know that the column of liquid rises by 5.25 cm when the temperature increases by 10.0 °F. So, we can set up a proportion: \(\frac{5.25 \mathrm{~cm}}{10.0^{\circ}\mathrm{F}} = \frac{18.5 \mathrm{~cm}}{x}\) We need to solve for x, which represents the temperature increase in Fahrenheit.
02

Solve the proportion

Cross-multiply and divide to find the value of x: \(x = \frac{10.0^{\circ}\mathrm{F} \times 18.5 \mathrm{~cm}}{5.25 \mathrm{~cm}} = \ 35.2381... \,^{\circ}\mathrm{F}\) Therefore, the temperature increases by approximately 35.24 °F.
03

Convert the initial temperature to Celsius

We need to convert the initial temperature of 98.6 °F to Celsius. The conversion formula is: \(C = \frac{5}{9}(F - 32)\) So, the initial temperature in Celsius is: \(C = \frac{5}{9}(98.6 - 32) = 36.78^{\circ}\mathrm{C}\)
04

Convert the temperature increase to Celsius

Now, we need to convert the temperature increase of 35.24 °F to Celsius. We can use the same conversion formula: \(C = \frac{5}{9}(35.24 - 32) = 1.69^{\circ}\mathrm{C}\) Therefore, the temperature increase in Celsius is approximately 1.69 °C.
05

Calculate the final temperature in Celsius

Finally, we can find the final temperature in Celsius by adding the initial temperature and the temperature increase: Final Temperature = Initial Temperature + Temperature Increase Final Temperature = \(36.78^{\circ}\mathrm{C} + 1.69^{\circ}\mathrm{C} = 38.47^{\circ}\mathrm{C}\) So, the final temperature of the liquid in Celsius is approximately \(38.47^{\circ}\mathrm{C}\).

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