The National Weather Service uses the following formula to...

The National Weather Service uses the following formula to calculate the wind chill: $$ W=35.74+0.6215 T_{a}-35.75 V^{0.16}+0.4275 T_{a} V^{0.16} $$ where $W$ is the wind chill temperature in ${ }^{\circ} \mathrm{F}, T_{a}$ is the air temperature in ${ }^{\circ} \mathrm{F},$ and $V$ is the wind speed in miles per hour. Note that $W$ is defined only for air temperatures at or lower than $50^{\circ} \mathrm{F}$ and wind speeds above 3 miles per hour. (a) Suppose the air temperature is $42^{\circ}$ and the wind speed is 7 miles per hour. Find the wind chill temperature. Round your answer to two decimal places. (b) Suppose the air temperature is $37^{\circ} \mathrm{F}$ and the wind chill temperature is $30^{\circ} \mathrm{F}$. Find the wind speed. Round your answer to two decimal places.

The National Weather Service uses the following formula to calculate the wind chill:
$$
W=35.74+0.6215 T_{a}-35.75 V^{0.16}+0.4275 T_{a} V^{0.16}
$$
where $W$ is the wind chill temperature in ${ }^{\circ} \mathrm{F}, T_{a}$ is the air temperature in ${ }^{\circ} \mathrm{F},$ and $V$ is the wind speed in miles per hour. Note that $W$ is defined only for air temperatures at or lower than $50^{\circ} \mathrm{F}$ and wind speeds above 3 miles per hour.
(a) Suppose the air temperature is $42^{\circ}$ and the wind speed is 7 miles per hour. Find the wind chill temperature. Round your answer to two decimal places.
(b) Suppose the air temperature is $37^{\circ} \mathrm{F}$ and the wind chill temperature is $30^{\circ} \mathrm{F}$. Find the wind speed. Round your answer to two decimal places.

Key Concepts

Wind Chill

Wind chill is a measure that reflects the combined effect of wind and cold temperatures on the human body. It is used to indicate how cold it actually feels, rather than simply the air temperature, and serves as an important factor in weather forecasting and public safety during cold weather conditions.

Mathematical Modeling

Mathematical modeling involves creating formulas or equations that represent real-world phenomena. In this context, the wind chill formula models the relationship between air temperature, wind speed, and the perceived temperature on the human body. Such models are essential in meteorology to predict conditions and inform the public.

Substitution in Equations

Substitution is an algebraic technique where known values are inserted into an equation to simplify the expression or to solve for an unknown variable. It is a fundamental method used in solving equations and evaluating mathematical models, ensuring that abstract formulas can yield practical numerical results.

Fractional Exponents

Fractional exponents represent both roots and powers in a compact form. In the wind chill formula, the exponent 0.16 indicates that the wind speed is raised to a fractional power, which is equivalent to taking a specific root and then a power, and is an important concept when dealing with non-linear relationships in various mathematical models.

Solving Equations for Unknowns

This concept involves manipulating equations algebraically to isolate and determine the value of an unknown variable. Whether by simplifying expressions, applying inverse operations, or rearranging terms, solving for an unknown is a core aspect of algebra that is vital in both pure mathematics and applied sciences.

Rounding and Precision

Rounding involves approximating a numeric value to a specified degree of accuracy, which is crucial in real-world applications where exact values are unnecessary or unobtainable. In many scientific calculations, including weather modeling, rounding to a defined number of decimal places helps communicate results in a standardized and practical way.

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