Two helicopters, A and B, 25 kilometres apart flying at the same altitude, receive an S.O.S. distress signal from a cap-sized boat (C). The angle of depression from helicopter A to the boat is 40° while the angle of depression from helicopter B to the boat is 30 °.
a) Calculate the distance each helicopter must travel to reach the cap-sized boat.
b) Find the height of the two helicopters
Can someone solve this for me please<33
Did you make a sketch?
If so, then you must have seen that angle C = 110°
You have all the angles plus one side, so the sine law can be used.
AC/sin30 = 25/sin110
AC = 25sin30/sin110 = appr 13.302
find BC in the same way.
Now that we have AC,
sin 40 = h/AC
h = ACsin40 = ...
I'm assuming that
(a) C is between A and B
(b) the choppers plan to fly horizontally to a point just above C.
If that's the case, then if we draw a diagram, and let
x = horizontal distance from A to C
h = altitude of choppers
then we have
h/x = tan30°
h/(25-x) = tan40°
x = 25tan40° / (tan30° + tan40°)
If you want the distance flying directly down to the boat on the water, then use the Pythagorean Theorem once you have h and x.
Sure! Let's first visualize the situation described:
We have two helicopters, A and B, located 25 kilometers apart from each other. Both helicopters are at the same altitude. They receive an SOS distress signal from a capsized boat labeled C. The angle of depression from helicopter A to the boat is 40°, and the angle of depression from helicopter B to the boat is 30°.
To solve this problem, we can use trigonometry and some basic geometry principles.
a) Calculate the distance each helicopter must travel to reach the capsized boat:
To find the distance each helicopter must travel, we need to find the distance between each helicopter and the boat.
Let's assume that the distance from helicopter A to the boat is x kilometers, and the distance from helicopter B to the boat is y kilometers.
Since both helicopters are flying at the same altitude, we can consider the situation as a right-angled triangle. The horizontal distance between the two helicopters is 25 kilometers, and the angles mentioned give us the vertical distances from the helicopter to the boat.
Using trigonometry, we can write the following equations:
x = tan(40°) * y
y = tan(30°) * x
Now we can solve these two equations simultaneously to find the values of x and y.
Using the equation x = tan(40°) * y, substitute the value of y from the equation y = tan(30°) * x:
x = tan(40°) * (tan(30°) * x)
Simplify the equation:
x = (tan(40°) * tan(30°)) * x
Divide both sides of the equation by (tan(40°) * tan(30°)) to isolate x:
1 = (tan(40°) * tan(30°))
Now we can calculate the value of (tan(40°) * tan(30°)) either using a scientific calculator or an online trigonometric calculator.
Resulting value ≈ 0.5
So we have:
1 = 0.5 * x
Divide both sides of the equation by 0.5:
2 = x
Therefore, the distance between helicopter A and the capsized boat is 2 kilometers.
Now, substitute this value of x into the equation y = tan(30°) * x:
y = tan(30°) * 2
Again, use a calculator to find the value of tan(30°) ≈ 0.5774:
y = 0.5774 * 2
y ≈ 1.1548
Therefore, the distance between helicopter B and the capsized boat is approximately 1.15 kilometers.
So, helicopter A must travel 2 kilometers, and helicopter B must travel 1.15 kilometers to reach the capsized boat.
b) Find the height of the two helicopters:
To find the height of each helicopter, we can use the tangent function with the angle of depression.
For helicopter A:
tan(40°) = Height_A / 2
Rearrange the equation to solve for Height_A:
Height_A = 2 * tan(40°)
Using a calculator, find the value of tan(40°) ≈ 0.8391:
Height_A = 2 * 0.8391
Height_A ≈ 1.6782
Therefore, the height of helicopter A is approximately 1.68 kilometers.
For helicopter B:
tan(30°) = Height_B / 1.15
Rearrange the equation to solve for Height_B:
Height_B = 1.15 * tan(30°)
Using a calculator, find the value of tan(30°) ≈ 0.5774:
Height_B = 1.15 * 0.5774
Height_B ≈ 0.6641
Therefore, the height of helicopter B is approximately 0.66 kilometers.
So, the height of helicopter A is approximately 1.68 kilometers, and the height of helicopter B is approximately 0.66 kilometers.
I hope this helps! Let me know if you have any further questions.