Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides 1100 m due east and then turns due north and travels another 1430 m before reaching the campground. The second cyclist starts out by heading due north for 1770 m and then turns and heads directly toward the campground. (a) At the turning point, how far is the second cyclist from the campground? (a) At the turning point, how far is the second cyclist from the campground?

To find the distance of the second cyclist from the campground at the turning point, we can use the Pythagorean theorem.

Let's break down the steps to calculate this distance:

1. Draw a diagram to visualize the situation.

In this case, draw a diagram with two perpendicular axes, where the x-axis represents the east-west direction and the y-axis represents the north-south direction. Mark the starting point of both cyclists as the origin (0,0), and then plot the distances they travel accordingly.

2. Calculate the distance traveled by the first cyclist.

The first cyclist rides 1100 m east, so their position is (1100, 0). Then, they turn north and travel 1430 m, so their final position is (1100, 1430).

3. Calculate the distance traveled by the second cyclist.

The second cyclist first heads north for 1770 m, so their position is (0, 1770). Then, they turn directly toward the campground, so we need to find the distance from their current position to the campground.

4. Use the Pythagorean theorem to find the distance of the second cyclist from the campground at the turning point.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the distance of the second cyclist from the campground at the turning point is the hypotenuse, and the distances along the x-axis and y-axis are the other two sides.

So, the distance at the turning point can be calculated as:

distance = sqrt((1100-0)^2 + (1430-1770)^2)

= sqrt(1100^2 + (-340)^2)

= sqrt(1210000 + 115600)

= sqrt(1325600)

≈ 1151.56 meters

Therefore, at the turning point, the second cyclist is approximately 1151.56 meters from the campground.

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