Question

A small radio transmitter broadcasts in a 55 mile radius. If you drive along a straight line from a city 73 miles north of the transmitter to a second city 77 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Answer 1

Explanation:

To determine the duration of the drive during which you will pick up a signal from the transmitter, we need to calculate the distance between the transmitter and the line of travel between the two cities.

Let's consider the distance between the transmitter and the line connecting the two cities as a straight line segment. This forms a right triangle with the transmitter at one vertex and the two cities at the other vertices. The side lengths of this triangle can be calculated using the Pythagorean theorem.

The distance between the transmitter and the line of travel is given by the hypotenuse of the right triangle. Using the Pythagorean theorem, we have:

Distance^2 = (73 miles)^2 + (77 miles)^2

Distance ≈ 104.182 miles

Since the radio transmitter has a broadcast radius of 55 miles, if the distance between the transmitter and the line of travel is greater than the broadcast radius, there will be no signal during the entire drive.

In this case, the distance between the transmitter and the line of travel (104.182 miles) is greater than the broadcast radius (55 miles). Therefore, you will not pick up a signal from the transmitter during any part of the drive.

Answer 2

To determine the distance along the drive during which you will pick up a signal from the transmitter, we can use basic geometry and the concept of overlapping circles.

The radio transmitter has a broadcast radius of 55 miles. We can visualize the situation by drawing a diagram. Let's label the transmitter as point T, the city 73 miles north as point N, and the city 77 miles east as point E.

We can draw two circles with radii of 55 miles centered at N and E, respectively. The overlapping region between these two circles represents the area where you will be within range of the transmitter.

Using the Pythagorean theorem, we can calculate the distance between points N and E:

Distance NE = √((73^2) + (77^2)) ≈ 104.13 miles

Now, let's determine the overlapping region. We need to find the area of the intersection between the two circles. This can be calculated using the formula for the area of a circular segment.

Area of overlapping region = r^2 * acos((d^2 + r^2 - R^2) / (2 * d * r)) + R^2 * acos((d^2 + R^2 - r^2) / (2 * d * R)) - 0.5 * sqrt((-d + r + R) * (d + r - R) * (d - r + R) * (d + r + R))

where:

r = 55 miles (radius of the smaller circle centered at N)

R = 55 miles (radius of the smaller circle centered at E)

d = Distance NE ≈ 104.13 miles

By plugging in the values into the formula and calculating the area, we find:

Area of overlapping region ≈ 3837.14 square miles

Therefore, during your drive from the city 73 miles north to the city 77 miles east of the transmitter, you will pick up a signal from the transmitter for approximately 3837.14 square miles of the drive.

Explanation: