https://file.aiquickdraw.com/mathbot/file/1730779114756lt35tura.jpg

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? 

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**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 \( \sqrt[n]{x^m} = x^{m/n} \).

Windchill temperatures can be calculated using the formula below, where C is the windchill, in degrees Fahrenheit, t is the air temperature, in degrees Fahrenheit, and v is the wind speed, in miles per hour. Use the formula to answer parts a and b.

C = 35.74 + 0.6215t - 35.74 * (4th root of v^25) + 0.4275t * (4th root of v^4)

a. Rewrite the equation for calculating windchill temperatures using rational exponents.
b. Use the form of the equation in part (a) and a calculator to find the windchill temperature, to the nearest degree, when the air temperature is 10°F and the wind speed is 40 miles per hour.

Explore the windchill temperature calculation with rational exponents. Learn how to rewrite the formula for windchill temperatures using rational exponents and find the windchill temperature for specific conditions. Suitable for students in Grades 10-12.

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