Let's go through each part of this problem.

Part (a): Rewriting with Rational Exponents

The given formula for windchill temperature is: [ C = 35.74 + 0.6215t - 35.74 \left( \sqrt[4]{v^{25}} \right) + 0.4275t \sqrt[4]{v^4} ]

To rewrite this with rational exponents, we need to convert the roots and powers into rational exponents.

1. For $\sqrt[4]{v^{25}}$, this is equivalent to $v^{25/4}$.
2. For $\sqrt[4]{v^4}$, this is equivalent to $v^{4/4} = v^1 = v$.

Thus, the equation becomes: $C = 35.74 + 0.6215t - 35.74 \cdot v^{25/4} + 0.4275t \cdot v$

Part (b): Calculating Windchill Temperature

Now, we use this rewritten formula to find the windchill temperature $C$ when:

• $t = 10^\circ F$
• $v = 40$ miles per hour

Substitute these values into the formula: $C = 35.74 + 0.6215(10) - 35.74 \cdot 40^{25/4} + 0.4275 \cdot 10 \cdot 40$

Step-by-Step Calculation

1. Calculate $0.6215 \cdot 10 = 6.215$.
2. Calculate $0.4275 \cdot 10 \cdot 40 = 171$.
3. Calculate $40^{25/4}$. This part is complex, so let’s approximate it:
   • $40^{25/4} \approx 63335.98$ (using a calculator or software for this step).
   • Then, $-35.74 \cdot 63335.98 \approx -2263486.95$.

Now, add all these components together: $C \approx 35.74 + 6.215 - 2263486.95 + 171$ $C \approx -2262274.995$

Rounding to the nearest degree, $C \approx -2262275^\circ F$.

Would you like further details on any step, or have additional questions?

Additional Related Questions:

1. How would the windchill temperature change if the wind speed was doubled?
2. What is the effect of increasing the air temperature on the windchill calculation?
3. Why is the exponent $25/4$ used for the wind speed term in this formula?
4. How would the windchill temperature change if we used Celsius instead of Fahrenheit?
5. How would the formula change if wind speed was measured in kilometers per hour?

Tip: When converting roots to exponents, remember that $\sqrt[n]{x^m} = x^{m/n}$.