Two helicopters flying at the same altitude are 2000m apart when they spot a life raft below. The raft is directly between the two helicopters. The angle of depression from one helicopter to the raft is 37° and the angle of depression from the other helicopter is 49°. Both helicopters are flying at 170km/h. How long will it take the closer aircraft to reach the raft?
A triangle is formed:
base = 2000 m.,
A = 180 - (37+49) = 94o,
B = 37o,
C = 49o,
Use Law of Sine to solve triangle:
sin94/2000 = sin37/b,
b = 1207 m. = 1.207 km,
sin94/2000 = sin49/c,
c = 1513 m.,
d = V*t,
t = d/V = 1.207/170 = 0.0071h = 25.6 s.
did you make your sketch?
On mine I let the vertical height between the raft and the line of flight be h
and the distance from the 49° helicopter be x km, so the other part is 2 - x km
using the two right-angled triangles,
tan 49° = h/x ---> h = xtan49
tan 37° = h/(2 - x) ----> h = (2-x)tan37
xtan49 = (2-x)tan37
xtan49 = 2tan37 - xtan37
x(tan49 + tan37) = 2tan37
take over
To solve this problem, we can break it down into two parts:
1. Finding the distance between the closer helicopter and the raft.
2. Calculating the time it takes for the helicopter to reach the raft.
Let's start by finding the distance between the closer helicopter and the raft.
Step 1: Draw a diagram
To visualize the problem, it's helpful to draw a diagram. Draw two lines to represent the helicopters, and mark the angle of depression for each helicopter. Place a dot to represent the life raft at the center.
Step 2: Define variables
Let's denote the distance between the closer helicopter and the raft as x.
Step 3: Use trigonometry
Since we have the angle of depression and the distance between the helicopters, we can use trigonometry to find the distance between the closer helicopter and the raft.
Using the tangent function, we have:
tan(37°) = x / 2000m
Rearranging the equation, we get:
x = tan(37°) * 2000m
Now, let's calculate the value of x.
Using a calculator, we find:
x ≈ 0.7536 * 2000m
x ≈ 1507.2m
Therefore, the distance between the closer helicopter and the raft is approximately 1507.2 meters.
Now, let's move on to calculating the time it takes for the helicopter to reach the raft.
Step 1: Convert the helicopter's speed to m/s
We need to convert the speed of the helicopter from km/h to m/s to be consistent with the distance we found earlier.
170 km/h * (1000m / 1km) * (1h / 3600s) = 47.2222 m/s (rounded to 4 decimal places)
Therefore, the speed of the helicopter is approximately 47.2222 m/s.
Step 2: Use the formula for time
Time = Distance / Speed
To calculate the time it takes for the closer helicopter to reach the raft, we divide the distance by the speed:
Time = 1507.2m / 47.2222 m/s
Using a calculator, we find:
Time ≈ 31.9020 seconds (rounded to 4 decimal places)
Therefore, it will take approximately 31.9020 seconds for the closer helicopter to reach the raft.