To determine during how much of the drive you will pick up a signal from the transmitter, you need to consider the distances involved.
1. The first city is 40 miles north of the transmitter, and the second city is 42 miles east of the transmitter. This forms a right triangle with the transmitter at the right angle.
2. Using the Pythagorean theorem (a^2 + b^2 = c^2), where a = 40 miles and b = 42 miles, you can find the distance from the first city to the second city (the hypotenuse).
3. Calculate the distance: \(40^2 + 42^2 = c^2\).
\(1600 + 1764 = c^2\).
\(3364 = c^2\).
\(c = \sqrt{3364}\).
\(c = 58\).
Therefore, the distance from the first city to the second city is 58 miles. During the entire drive from the first city to the second city, you will pick up a signal from the transmitter, as the distance of 58 miles falls within the 35-mile radius of the transmitter's broadcast.
