You observe a plane approaching overhead at a speed of 600 mph. The angle of elevation of the plane is 14 degrees at one time and 54 degrees two min later.
A) What is the altitude of the plane?
B) how far from the plane are you (after two minutes)?
I drew a side-view of the situation.
Le your position on the ground be C
Let the first position of the plane be A
let the new position of the plane 2 minutes later be B
Let P be the point on the ground directly below A
given: angle PCA = 14°= angle CAB (alternate angles with parallel lines)
angle PCB = 54° which makes angle ACB = 40°
in 2 minutes, the plane went 600(2/6) = 20 miles
so AB = 20
Also angle B = 180-40-14 = 126°
by the Sine Law:
BC/sin14 = 20/sin126
BC = 5.98 miles
The height :
drop a perpendicular from B to the base CP, let it be h
We have a right-angled triangle.
Sin 54° = h/5.98
h = 5.98sin54 = 4.838 or appr 4.8 miles high
To find the altitude of the plane and the distance between you and the plane, we can use trigonometry and basic geometry principles.
Let's start by drawing a diagram. Consider a right triangle where the side opposite the 14-degree angle represents the altitude of the plane (h), and the side opposite the 54-degree angle represents the horizontal distance between you and the plane (d).
Now, we can use the tangent function, which relates the opposite side to the adjacent side of a right triangle, to find the altitude of the plane.
A) Altitude of the plane (h):
At the 14-degree angle, the tangent of the angle (tan(14)) is equal to the altitude divided by the horizontal distance (d).
tan(14) = h / d
Since we don't know the value of d, we need to eliminate it to solve for the altitude. We can use the fact that the plane is moving at a constant speed of 600 mph. In two minutes, which is 1/30th of an hour, the plane will have traveled a distance of 600 / 30 = 20 miles.
Therefore, after two minutes, the horizontal distance (d) is 20 miles, or 20 * 5280 feet.
Now we can substitute this value into our equation:
tan(14) = h / (20 * 5280)
To solve for h, we multiply both sides of the equation by (20 * 5280):
h = (20 * 5280) * tan(14)
Using a calculator, we find that h ≈ 2,097 feet (rounded to the nearest foot). Therefore, the altitude of the plane is approximately 2,097 feet.
B) Distance between you and the plane (d):
We have already determined that after two minutes, the horizontal distance (d) is 20 miles, or 20 * 5280 feet. So, the distance between you and the plane is approximately 20 * 5280 = 105,600 feet.
Therefore, after two minutes, you are approximately 105,600 feet away from the plane.