A rocket is launched vertically into the air and is observed from a tower that is 1.5 km above the ground level. Soon after the launch the rocket is at an angle of 25 degrees. Later the rocket has climbed vertically a further 4 km and its angle of elevation from the tower is 66 degrees. How far is the observer from the launched site? Assuming that the base of the tower and the launched site is on a perfectly horizontal ground level.
measuring from the tower, if the distance away is x, and the rocket had risen to a height h when first observed,
h/x = tan25°
(h+4)/x = tan66°
Now you can solve for x.
To solve this problem, we can use basic trigonometry and the concept of similar triangles. Let's break it down step by step:
1. Draw a diagram: First, let's draw a diagram to visualize the problem. We have a tower, the rocket, and the observer. The observer is located on top of the tower, which is 1.5 km above the ground level. The angle of elevation from the observer to the rocket is 25 degrees initially and 66 degrees after the rocket has climbed further.
2. Identify the relevant triangles: We have two triangles in this problem - the smaller triangle formed by the observer, the tower, and the rocket, and the larger triangle formed by the observer, the tower, and the launched site. Let's label the tower's height as h and the distance between the observer and the launched site as x (which we're trying to find).
3. Solve for the small triangle: In the smaller triangle, we have the opposite side (vertical distance) and the angle of elevation. We can use the tangent function to find the distance between the observer and the rocket when it is at an angle of 25 degrees. Let's call this distance d1.
tan(25 degrees) = h / d1
d1 = h / tan(25 degrees)
4. Solve for the larger triangle: In the larger triangle, we know the height of the tower is h + 1.5 km and the angle of elevation is 66 degrees. We can use the tangent function again to find the distance between the observer and the launched site. Let's call this distance d2.
tan(66 degrees) = (h + 1.5 km) / d2
d2 = (h + 1.5 km) / tan(66 degrees)
5. Set up an equation: Since the observer is at the same horizontal distance from both the rocket and the launched site, we can equate d1 and d2:
d1 = d2
h / tan(25 degrees) = (h + 1.5 km) / tan(66 degrees)
6. Solve the equation: Now, we can solve the equation for h, which will give us the height of the tower:
h = [(h + 1.5 km) / tan(66 degrees)] * tan(25 degrees)
Simplifying the equation further, we get:
(h * tan(25 degrees)) - [(h + 1.5 km) / tan(66 degrees)] = 0
Solve this equation for h using algebraic methods or numerical approximation methods. Once we find h, we can substitute it back into the equation d2 = (h + 1.5 km) / tan(66 degrees) to find x, the distance between the observer and the launched site.
7. Calculate the final answer: Once you find the value of x, you will have the distance between the observer and the launched site.