In the Fahrenheit temperature scale, water freezes at 32^∘ F and boils at 212^∘ F. In the Celsiu

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In the Fahrenheit temperature scale, water freezes at 32^∘ F...

In the Fahrenheit temperature scale, water freezes at $32^{\circ} \mathrm{F}$ and boils at $212^{\circ} \mathrm{F}$. In the Celsius scale, water freezes at $0^{\circ} \mathrm{C}$ and boils at $100^{\circ} \mathrm{C}$. Assuming that the Fahrenheit temperature $F$ and the Celsius temperature $C$ are related by a linear equation, find $F$ in terms of $C$. Use your equation to find the Fahrenheit temperatures corresponding to $30^{\circ} \mathrm{C}, 22^{\circ} \mathrm{C},-10^{\circ} \mathrm{C}$, and $-14^{\circ} \mathrm{C}$, to the nearest degree.

Key Concepts

Linear Relationships

A linear relationship is one in which two variables change at a constant rate relative to each other. In the context of converting one measurement unit to another, such as converting temperatures between scales, this constant rate is expressed as a straight line when the variables are graphed. The simplicity of linear relationships makes them useful for modeling and conversion problems.

Slope and Intercept

In a linear equation, the slope represents the rate of change between two variables, while the intercept is the value of the function when the independent variable is zero. When converting between temperature scales, the slope accounts for the size of each unit increment, and the intercept adjusts the starting point of the scale. Understanding these components is essential for formulating a conversion formula.

Temperature Scale Conversion

Temperature scale conversion involves finding a mathematical relationship between two different temperature systems, such as Celsius and Fahrenheit. This requires identifying fixed calibration points on both scales—typically the freezing and boiling points of water—and then deriving the corresponding linear equation to convert any temperature from one scale to the other.

Calibration Using Reference Points

Calibration using reference points involves using known values—in this case, the temperatures at which water freezes and boils on both temperature scales—to determine the parameters of the conversion equation. By assigning these known values to the linear equation, one can solve for the slope and intercept, ensuring the conversion formula accurately reflects the relationship between the scales.

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