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Chapter 10: Problem 34

At what temperature, the Fahrenheit and the Celsius scales will give numerically equal (but opposite in sign) values? (A) \(-40^{\circ} \mathrm{F}\) and \(40^{\circ} \mathrm{C}\) (B) \(11.43^{\circ} \mathrm{F}\) and \(-11.43^{\circ} \mathrm{C}\) (C) \(-11.43^{\circ} \mathrm{F}\) and \(+11.43^{\circ} \mathrm{C}\) (D) \(+40^{\circ} \mathrm{F}\) and \(-40^{\circ} \mathrm{C}\)

Short Answer

Expert verified
The Fahrenheit and Celsius scales will give numerically equal (but opposite in sign) values at \(-40^{\circ}\). Therefore, choice (D) \(+40^{\circ} \mathrm{F}\) and \(-40^{\circ} \mathrm{C}\) is the correct answer.

Step by step solution

01

Understand the conversion formula

The formula that converts Celsius to Fahrenheit is given by \( F = \frac{9}{5}C + 32 \), where \( F \) is the temperature in Fahrenheit and \( C \) is the temperature in Celsius.
02

Substitute the given equal relationship

We know from the problem statement that we are looking when \(F\) and \(C\) are numerically equal but opposite in sign, so we can write this as \(F = -C\). Substitute \( C = -F \) into the conversion formula.
03

Solve the equation

Substituting into the conversion formula yields \( F = \frac{9}{5}(-F) + 32 \). Solve this equation by first distributing the \(\frac{9}{5}\) to get \( F = -\frac{9}{5}F + 32 \). Then combine like terms by adding \(\frac{9}{5}F\) to both sides of the equation to get \( \frac{14}{5}F = 32 \). Finally, solve for \( F \) by multiplying both sides of the equation by \( \frac{5}{14} \). This gives \( F = -40 \). Therefore, the Fahrenheit and Celsius scales will give numerically equal but opposite in sign values at -40 degrees.

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Most popular questions from this chapter

The amount of heat required will be minimum when a body is heated through (A) \(1 \mathrm{~K}\) (B) \(1^{\circ} \mathrm{C}\) (C) \(1^{\circ} \mathrm{F}\) (D) It will be the same in all the three cases

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