Problem 3 - Calculus for Biology and Medicine

Question

In cold weather the temperature that we feel depends both on the actual air temperature and on whether the wind is blowing or not. Strong winds blow heat away from our bodies, increasing their rate of heat loss and making us feel colder, a phenomenon known as wind chill. In cold winters, weather forecasters report both the real temperature and the apparent temperature including wind chill. One widely used formula for the apparent temperature \(W\), when the air temperature (measured in \(\left.{ }^{\circ} \mathrm{F}\right)\) is \(T\), and the wind speed (measured in mph) is \(V\), is: $$W=35.74+0.6215 T-35.75 V^{0.16}+0.4275 T V^{0.16}$$ This formula is only accurate when \(T \leq 50^{\circ} \mathrm{F}\) and \(V \geq 3 \mathrm{mph}\). (a) Use \((10.2)\) to calculate the apparent temperatures felt by a person in each of the three following locations: (i) Boston in January \(\left(T=14^{\circ} \mathrm{F}, V=11 \mathrm{mph}\right)\) (ii) Minneapolis in January \(\left(T=23^{\circ} \mathrm{F}, V=13 \mathrm{mph}\right)\) (iii) Chicago in January \(\left(T=24^{\circ} \mathrm{F}, V=18 \mathrm{mph}\right)\) In which location will a person feel coldest? (b) In this part we will see the danger of trying to evaluate a function outside of its correct domain. Calculate the wind chill for Los Angeles in January \(\left(T=65^{\circ} \mathrm{F}, V=0.8 \mathrm{mph}\right) .\) [Note that these \((T, V)\) values are outside the domain for \(W(T, V) .]\) Does your answer make sense?

Step-by-Step Solution

Answer

Boston feels the coldest with a wind chill of -4.50°F. The calculation for Los Angeles is not valid.

Step 1: Calculate Wind Chill for Boston

We need to calculate the apparent temperature \(W\) for Boston using the formula:\[W = 35.74 + 0.6215T - 35.75V^{0.16} + 0.4275TV^{0.16}\]For Boston, \(T = 14^\circ F\) and \(V = 11 \, mph\). Substitute these values:\[W = 35.74 + 0.6215 \times 14 - 35.75 \times 11^{0.16} + 0.4275 \times 14 \times 11^{0.16}\]Calculate each term step by step:- \(0.6215 \times 14 = 8.701\)- \(11^{0.16} \approx 1.644\)- \(35.75 \times 1.644 = 58.785\)- \(0.4275 \times 14 \times 1.644 \approx 9.84\)Now substitute back into the formula:\[W \approx 35.74 + 8.701 - 58.785 + 9.84 \approx -4.50^\circ F\]

Step 2: Calculate Wind Chill for Minneapolis

Substitute Minneapolis values, \(T = 23^\circ F\) and \(V = 13 \, mph\):\[W = 35.74 + 0.6215 \times 23 - 35.75 \times 13^{0.16} + 0.4275 \times 23 \times 13^{0.16}\]- Calculate \(0.6215 \times 23 = 14.3045\)- \(13^{0.16} \approx 1.732\)- \(35.75 \times 1.732 = 61.945\)- \(0.4275 \times 23 \times 1.732 \approx 16.997\)Substitute back into the formula:\[W \approx 35.74 + 14.3045 - 61.945 + 16.997 \approx 5.10^\circ F\]

Step 3: Calculate Wind Chill for Chicago

Substitute Chicago values, \(T = 24^\circ F\) and \(V = 18 \, mph\):\[W = 35.74 + 0.6215 \times 24 - 35.75 \times 18^{0.16} + 0.4275 \times 24 \times 18^{0.16}\]- Calculate \(0.6215 \times 24 = 14.916\)- \(18^{0.16} \approx 2.045\)- \(35.75 \times 2.045 = 73.09875\)- \(0.4275 \times 24 \times 2.045 \approx 20.9778\)Substitute back into the formula:\[W \approx 35.74 + 14.916 - 73.09875 + 20.9778 \approx -1.465^\circ F\]

Step 4: Identify Coldest Location

Based on the calculated wind chill temperatures:- Boston: \(-4.50^\circ F\)- Minneapolis: \(5.10^\circ F\)- Chicago: \(-1.465^\circ F\)Boston is the coldest location with the lowest apparent temperature of \(-4.50^\circ F\).

Step 5: Calculate Wind Chill for Los Angeles

Check if the calculation is valid: \(T = 65^\circ F\) and \(V = 0.8 \, mph\) are outside the domain. However, calculate it to see consequences:\[W = 35.74 + 0.6215 \times 65 - 35.75 \times 0.8^{0.16} + 0.4275 \times 65 \times 0.8^{0.16}\]- Calculate \(0.6215 \times 65 = 40.3975\)- \(0.8^{0.16} \approx 0.955\)- \(35.75 \times 0.955 = 34.11625\)- \(0.4275 \times 65 \times 0.955 \approx 26.4878\)Substitute back into the formula:\[W \approx 35.74 + 40.3975 - 34.11625 + 26.4878 \approx 68.50905^\circ F\]

Step 6: Evaluate Result Validity

Your calculation for Los Angeles gives \(68.50905^\circ F\), which is not physically meaningful as the temperature is above the domain \(T \leq 50^\circ F, V \geq 3 \, mph\). The formula should not be used in these conditions and results might be misleading.

Key Concepts

Apparent Temperature

In chilly weather, the apparent temperature is what actually impacts how we feel. It's influenced by both the real temperature and how fast the wind is blowing. Just hearing a temperature reading isn't enough. Wind speed can increase the rate at which our bodies lose heat, making us experience cooler temperatures than what the thermometer shows. This perceived coolness due to the wind is called wind chill. Weather reports often include this apparent temperature to give a more accurate sense of what you'll feel when you step outside.

Wind Speed

Wind speed is a crucial factor in calculating wind chill, which affects the apparent temperature. The stronger the wind, the more it enhances heat loss from your body. When the wind blows, it strips away the layer of warm air close to your skin, making you feel colder. This is why, on windy days, a lower temperature may feel colder than it actually is. The wind speed in the wind chill formula is measured in miles per hour (mph), and it is a significant variable that changes how cold the air feels against your skin.

Domain of Function

The wind chill formula has specific conditions where it is designed to work, known as its domain. This formula is intended only for air temperatures of 50°F or below and wind speeds of at least 3 mph. This means if the temperature is above 50°F or the wind speed is less than 3 mph, the formula may not provide accurate results. For example, in warmer climates like Los Angeles in January, using the formula outside its domain will give unreliable results, showing a misleading apparent temperature. It's essential to use the formula strictly within its specified range for correct wind chill values.

Calculation Steps

To find the wind chill and thus the apparent temperature, you need to follow a few steps using the wind chill formula:

• First, identify the actual temperature (\(T\)) and the wind speed (\(V\)).
• Substitute these values into the formula \[ W = 35.74 + 0.6215T - 35.75V^{0.16} + 0.4275TV^{0.16} \]
• Perform the calculations by multiplying terms inside the formula.
• Solve each part step-by-step, and then combine them to find the final wind chill value (W).

By carefully carrying out these steps, you can accurately determine how cold air will feel on a given windy, cold day.