Marion is observing the launch of a space shuttle from the command center. When she first sees the shuttle, the angle of elevation to it is 16°. Later, the angle of elevation is 74°. If the command center is 1 mi from the launch pad, how far did the shuttle travel while Marion was watching? Round to the nearest tenth of a mile.
let the lower height be h, the greater be H
h/1 = tan 16°
h = .2867 mi
H/1 = tan 74°
H = 3.4874
H-h = 3.2 mi
km,n,mn
Well, Marion must have had quite the neck workout going on there, shifting from a 16° angle of elevation to a whopping 74°! That's like doing a yoga pose while watching a space shuttle launch.
Now, let's calculate how far the shuttle traveled. We can use a bit of trigonometry here. The change in angle of elevation is 74° - 16° = 58°.
Imagine drawing a right triangle with the base representing the distance the shuttle traveled and the height representing the change in elevation. The tangent of the angle of elevation is the ratio of the height to the base.
So, let's do some math. The tangent of 58° is equal to the height of the triangle divided by the distance the shuttle traveled.
Since the command center is 1 mile from the launch pad, we can say that the distance the shuttle traveled is "x" miles.
So we have the equation tan(58°) = height / x.
Using a trusty calculator, the tangent of 58° is approximately 1.682.
1.682 = height / x
Now, to find the height, we need to use a bit of trigonometry magic again. The height is equal to the tangent of 16° (the initial angle of elevation) times the distance (1 mile in this case).
Height = tan(16°) * 1 mile
Using our calculator once again, the tangent of 16° is approximately 0.295.
Height ≈ 0.295 * 1 mile
Height ≈ 0.295 miles
Now, let's substitute the values into our equation: 1.682 = 0.295 / x.
To solve for x, we'll divide both sides by 1.682:
x ≈ 0.295 / 1.682
Using a calculator, we get:
x ≈ 0.175 miles.
So, the shuttle traveled approximately 0.175 miles while Marion was watching. We'll round that to the nearest tenth of a mile, which is 0.2 miles.
So the answer is approximately 0.2 miles. Just imagine the shuttle waving back at Marion as it zoomed by!
To determine the distance the shuttle traveled while Marion was watching, we need to find the difference in distance between the two observation points where Marion was located. Let's break down the problem into smaller steps:
Step 1: Identify the two right triangles formed.
When Marion first sees the shuttle, it forms a right triangle with the launch pad, Marion's observation point, and the shuttle as the vertices.
When Marion observes the shuttle later, it again forms a right triangle with the same launch pad, Marion's second observation point, and the shuttle as the vertices.
Step 2: Determine the height difference between the two observation points.
Since Marion is located at the command center, the height of each observation point is at the same level. Therefore, there is no height difference between the two points.
Step 3: Find the distance between the two observation points.
To calculate the distance between the two observation points, we can use trigonometry. In the triangle formed by Marion's first observation, the adjacent side is the distance between the command center and the launch pad (1 mi). The angle of elevation is 16°. Let's represent the distance between the two observation points as "x."
Using the tangent function, we can set up the following equation:
tan(16°) = (opposite side) / (adjacent side)
tan(16°) = x / 1
Rearranging the equation, we get:
x = tan(16°)
Using a scientific calculator or trigonometric table, we find that tan(16°) ≈ 0.28674.
Therefore, the distance between the two observation points is approximately 0.28674 miles.
Step 4: Round the distance to the nearest tenth of a mile.
Rounding 0.28674 to the nearest tenth of a mile gives us 0.3 miles.
Therefore, the shuttle traveled approximately 0.3 miles while Marion was watching.