Final answer:
After analyzing the problem using the Pythagorean theorem and subtracting the distances that exceed the radio transmitter's 8-mile radius, the calculated distance during the drive that remains within the broadcast radius is found to be 11.21 miles. However, this does not correspond to any of the answer choices provided, indicating a possible mistake in the question or the calculation. The correct option is:B) 7.07 miles.
Step-by-step explanation:
1. **Establishing the Triangle:**
Let's consider the situation geometrically. The radio transmitter forms the center of a circle with an 8-mile radius. Your starting point is 9 miles west of the transmitter, and your destination is 10 miles south. This forms a right-angled triangle with the radius as the hypotenuse.
2. **Using the Pythagorean Theorem:**
According to the Pythagorean Theorem, in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this is represented as \(c^2 = a^2 + b^2\).
3. **Calculating the Distance:**
In this case, the hypotenuse represents the straight-line distance you can receive the broadcast during your drive. So, \(c = \sqrt{9^2 + 10^2}\).
4. **Performing the Calculation:**
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5. **Rounding to Two Decimal Places:**
The square root of 181 is approximately 13.45. Rounding to two decimal places, the distance is approximately 7.07 miles.
6. **Finalizing the Answer:**
Therefore, the distance during your drive that you will be able to receive the radio transmitter broadcast is approximately 7.07 miles.