Question

The formula used by the National Weather Service to calculate wind chill in degrees Fahrenheit is: $$ 35.74+0.6215 \times T-35.75 \times\left(V^{(0.16)}\right)+0.4275 \times T \times\left(V^{(0.16)}\right) $$ \(T\) represents the air temperature in degrees Fahrenheit, and \(V\) represents the wind velocity in miles per hour. Which of these is an accurate statement about the relationship between wind chill and temperature? I. The relative impact on wind chill of a particular increase in wind speed is more significant at lower wind speeds than at higher wind speeds. II. Wind chill has an impact on the relative temperature feeling only at temperatures greater than or equal to \(35.75\) degrees Fahrenheit. II. Wind chill and temperature are inversely related to one another. (A) I only (B) II only (C) \(\mid\) and \(\|\) only (D) \(\|\) and IIII only

Short answer

The correct answer is (A) I only.

Step-by-step solution

Analyzing the Formula

The given formula for wind chill is \( 35.74 + 0.6215 \times T - 35.75 \times (V^{0.16}) + 0.4275 \times T \times (V^{0.16}) \). In this formula, \( T \) is the air temperature, and \( V \) is the wind speed. The term \( -35.75 \times (V^{0.16}) + 0.4275 \times T \times (V^{0.16}) \) indicates that wind speed and temperature impact wind chill.

Evaluating Statement I

Statement I claims the impact on wind chill from an increase in wind speed is more significant at lower speeds. The term \( V^{0.16} \) grows slower as \( V \) increases due to the power being less than 1, which means changes in wind speed result in larger proportional changes at lower speeds, confirming that statement I is accurate.

Evaluating Statement II

Statement II suggests wind chill affects the perceived temperature only for \( T \geq 35.75 \) degrees Fahrenheit. However, the wind chill formula helps calculate a lower perceived temperature primarily at lower air temperatures, not 35.75, making this statement inaccurate.

Evaluating Statement III

Statement III suggests wind chill and temperature are inversely related. As \( T \) increases, the perceived temperature or wind chill approaches \( T \); as \( T \) decreases, the wind chill will decrease. Thus, it initially seems they may be inversely related, but the direct temperature's magnitude doesn't decrease wind chill, making this statement inaccurate.

Conclusion

After examining the statements, statement I is the only accurate one. Therefore, the answer to the problem is option (A), as only statement I is true.

Key concepts

Understanding the Wind Chill Formula

The wind chill formula is a critical mathematical representation used to measure how cold the air feels on the skin. This perception is relative to the actual air temperature due to the effects of wind speed. The formula given by the National Weather Service for calculating wind chill in degrees Fahrenheit is:
\[35.74 + 0.6215 \times T - 35.75 \times (V^{0.16}) + 0.4275 \times T \times (V^{0.16})\]
Here, \( T \) stands for the actual air temperature in degrees Fahrenheit, and \( V \) represents the wind velocity in miles per hour. Breaking down the formula:

• The constant \( 35.74 \) provides a baseline temperature.
• \( 0.6215 \times T \) adjusts this baseline depending on the air temperature.
• \(-35.75 \times (V^{0.16})\) shows how wind speed reduces the temperature perception.
• The term \( 0.4275 \times T \times (V^{0.16})\) indicates how both temperature and wind speed interact to influence perceived cold.

Understanding the wind chill formula helps us grasp why colder temperatures feel even colder in windy conditions. The elements of the formula each add or subtract based on the conditions to give us a practical measure for preparing against the cold.

Temperature and Wind Speed Relationship in Wind Chill

The wind chill formula illustrates a fascinating relationship between temperature and wind speed. This relationship primarily answers how a particular condition of wind speed affects the perceived temperature. Here's how it works:

• The expression \( V^{0.16} \) indicates how wind speed contributes to the wind chill calculation.
• The exponent \(0.16\) is key—it means the rate at which wind impacts perceived temperature diminishes as speed increases. Essentially, initial wind increases have a more pronounced effect than when the wind is already strong.
• This effect is particularly noticeable when wind speeds are low. The lower the wind speed, the more significant any increase will feel.
• Consequently, the perception of colder weather grows less noticeably with further increases in already high wind speeds.

This progressive relationship illustrates why the impact of a small increase in wind speed is more detected at lower speeds, validating statement I in our original exercise. Wind chill helps individuals better prepare for outdoor activities by anticipating how cold it will feel under various conditions.

PSAT Math Problem Solving Skills

The PSAT Math section focuses significantly on problem-solving skills, often involving practical scenarios like the wind chill formula to test understanding. The problem presents several statements regarding the nature of wind chill, requiring careful analysis:

• PSAT problems often involve worded statements which call for converting real-world scenarios into mathematical insights.
• For instance, Statement I was judged true because of understanding how changes in wind speed affect wind chill, a process backed by analyzing the formula itself.
• The inaccurate statements II and III serve to enhance testing vigilance, as they could mislead without proper comprehension.

In this problem, the thoughtful breakdown of mathematical language and contextual clues ensures that the problem-solver is not merely guessing but making informed decisions. This aligns with PSAT objectives where students sharpen skills through logical reasoning and application of mathematical concepts. Understanding the language and nuances of such formulas enables students to approach PSAT math questions confidently and accurately.