Using data for the effect of temperature and wind on an exposed face, the National
Weather Service uses the following formula,
Wind Chill Index = 35.74 + 0.6215𝑇 − 35.75𝑉!.!" + 0.4275𝑇𝑉!.!"
𝑇 is the temperature in degrees Fahrenheit and 𝑉 is the velocity of the wind in miles per hour. Frostbite
occurs in about fifteen minutes when the wind chill index is about −20. Find the wind speed that produces
a wind chill index of −20 when the temperature is 5℉.
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To find the wind speed that produces a wind chill index of -20 when the temperature is 5°F, we need to use the given formula for the wind chill index and solve for the wind speed (V).
The formula for the wind chill index is:
Wind Chill Index = 35.74 + 0.6215T - 35.75V^(0.16) + 0.4275TV^(0.16)
Given:
Wind Chill Index = -20
Temperature (T) = 5°F
We can substitute the given values into the formula and solve for V.
-20 = 35.74 + 0.6215(5) - 35.75V^(0.16) + 0.4275(5)V^(0.16)
Simplifying the equation:
-55.74 = -35.75V^(0.16) + 2.1375V^(0.16)
Let's perform a numerical approximation to solve for V using an iterative method.
1. Start with an initial guess for V, let's say V = 10 mph.
2. Substitute this value of V in the equation:
-55.74 = -35.75(10)^(0.16) + 2.1375(10)^(0.16)
3. Calculate the new value of V based on the equation:
V = (0.0911V + 55.74)^(1/0.16)
4. Repeat steps 2 and 3 with the newly calculated value of V, until the value of V converges (reaches a stable value).
5. Once V converges, the resulting value will be the wind speed that produces a wind chill index of -20 when the temperature is 5°F.