Gloria is planning to ride her bicycle near the transmitter of a radio station. She will be

riding from a point 10 miles due east of the transmitter on a straight road to a point 40 miles
west and 10 miles north of the transmitter. She has a radio on her bicycle which can pick
up the station as long as she is within 10 miles of it. If she rides at a constant speed of 16 miles per hour, for what length of time during her ride will she be able to listen to the radio station?

To find the length of time during Gloria's ride that she will be able to listen to the radio station, we need to determine the distance she covers while being within a 10-mile radius of the radio station.

First, let's visualize Gloria's ride on a map:

^ N
|
|
|
|10 miles
| /
| /
T | / 40 miles
R |/_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
A | |
N | |
S | |
M | |10 miles
I | |
T | 10 miles |
T |—————————————————————————————————————————————————————————————

Here, the radio station is denoted by "T" and Gloria's ride starts 10 miles east of the transmitter and ends 40 miles west and 10 miles north of it.

To calculate the distance Gloria covers within a 10-mile radius, we need to find the length of the hypotenuse of the triangle formed by the radio station and Gloria's starting and ending points. We can use the Pythagorean theorem for this purpose.

The distance traveled is the square root of ((40 + 10)^2 + 10^2).

Calculating this, we get:

sqrt((50)^2 + 10^2)
= sqrt(2500 + 100)
= sqrt(2600)
≈ 50.99 miles

Now, we can determine the time it takes for Gloria to cover this distance by using the formula:

time = distance / speed

Plugging in the values:

time = 50.99 miles / 16 miles per hour
≈ 3.1875 hours

Therefore, Gloria will be able to listen to the radio station for approximately 3.1875 hours during her ride.

To determine the length of time Gloria will be able to listen to the radio station during her ride, we need to calculate the distance she will be from the transmitter at any given point.

Let's break down Gloria's ride into two parts:
1. From the point 10 miles due east of the transmitter to the transmitter itself.
2. From the transmitter to the point 40 miles west and 10 miles north of the transmitter.

First, let's calculate the distance Gloria will be from the transmitter during the first part of her ride. She starts 10 miles due east of the transmitter, and her final destination is the transmitter itself. This forms a right-angled triangle.

Using the Pythagorean theorem, we can calculate the distance as follows:
Distance = sqrt((10^2) + (10^2)) = sqrt(200) ≈ 14.1 miles

Since Gloria's radio can pick up the station as long as she is within 10 miles of it, during the first part of her ride, she will be within range for the entire duration.

Next, let's calculate the distance Gloria will be from the transmitter during the second part of her ride. She starts at the transmitter, and her final destination is a point 40 miles west and 10 miles north of the transmitter. This also forms a right-angled triangle.

Again, using the Pythagorean theorem, we can calculate the distance as follows:
Distance = sqrt((40^2) + (10^2)) = sqrt(1700) ≈ 41.2 miles

Since this distance is greater than the range of Gloria's radio (10 miles), she will not be able to listen to the radio station during this part of her ride.

To calculate the time during which Gloria will be able to listen to the radio station, we need to determine how long it takes her to complete the first part of her ride. To do this, we can use the formula:

Time = Distance / Speed

Time = 14.1 miles / 16 miles per hour ≈ 0.88125 hours or approximately 52.9 minutes.

Therefore, Gloria will be able to listen to the radio station for approximately 52.9 minutes during her ride.