On a clear day, you can see 11.5 km in any direction from a particular lookout. If a straight road is 6.9 km from the lookout, what length of the road can be seen from the lookout?
Draw a diagram. You have a circle of radius 11.5, and a chord 6.9 from the center.
If the length of road is 2x, then
6.9^2 + x^2 = 11.5^2
find x, then 2x
4.6 km
To find the length of the road that can be seen from the lookout, we need to determine how much of the road falls within the maximum visible range of 11.5 km.
Since the lookout provides visibility in any direction, we can visualize a circle with a radius of 11.5 km centered at the lookout. This circle represents the extent of what can be seen from the lookout.
Now, let's imagine drawing a perpendicular line from the lookout to the road. This line represents the shortest distance between the lookout and the road.
Since the road is 6.9 km away from the lookout, we can draw a line segment from the center of the circle to a point on the circle that is 6.9 km away. This line segment would be perpendicular to the road.
By drawing this line segment, we can see that it intersects the circle at two distinct points. The length of the road that can be seen from the lookout is given by the distance between those two points.
To calculate this length, we need to find the length of the line segment between the two points of intersection. This can be done using basic geometry and the properties of right-angled triangles.
First, we find the length of the line segment connecting the lookout to the point on the circle closest to the road. This line segment can be viewed as the hypotenuse of the right-angled triangle formed by the line segment, the radius of the circle (11.5 km), and the portion of the radius from the center to the point of intersection (6.9 km).
Using the Pythagorean theorem, we can calculate this length as follows:
length = √(radius^2 - portion^2)
= √(11.5^2 - 6.9^2)
≈ √(132.25 - 47.61)
≈ √84.64
≈ 9.2 km (rounded to one decimal place)
Therefore, the length of the road that can be seen from the lookout is approximately 9.2 km.