A jet A is flying above 2 ships B and C which are 18km apart. If the angles of depression of B and C from are 60 and 32 degrees respectively, find the height of the jet above the sea level, correct to the nearest m.
How good are you at sketching 3-D diagrams?
I let P be the point directly below plane A,
and I now have 4 triangles, each in their own plane.
Redraw each with a side view:
right-triangle ABP , right angle at P
right-triangle ACP, right angle at P
triangle BCP , (on ocean floor)
triangle ABC
let AP, the height of the plane be h km
in triangle ABP , sin60 = h/BA
h = ABsin60
in triangle ACP , sin 32 = h/AC
h = ACsin32
therefore, ABsin60 = ACsin32
√3/2AB = .5299AC
AB = .6119 AC
See if that gets you anywhere.
I hope I am not overthinking this problem
To solve this problem, we can use trigonometry and the concept of similar triangles. Here are the steps to find the height of the jet above the sea level:
Step 1: Draw a diagram representing the situation described in the problem. Label the jet as point J, ship B as point B, and ship C as point C. Draw a straight line segment connecting ships B and C and label it as line segment BC. Also, draw a vertical line from point J and label it as the height of the jet (h).
Step 2: Since we're dealing with angles of depression, the angle between the horizontal line and line segment BC at point B is 90 degrees minus the angle of depression at point B, which is 90 - 60 = 30 degrees. Similarly, the angle between the horizontal line and line segment BC at point C is 90 - 32 = 58 degrees.
Step 3: Now, we have a right triangle JBC formed by the vertical line (h) and the line segment BC. The opposite side of the angle at point B is h and the adjacent side is BC. The opposite side of the angle at point C is also h, but the adjacent side is the remaining length of BC, which is 18 km.
Step 4: Applying the tangent function, we can write the following equations:
tan(30 degrees) = h / BC
tan(58 degrees) = h / (18 - BC)
Step 5: Rearrange the equations to solve for h:
h = BC * tan(30 degrees)
h = (18 - BC) * tan(58 degrees)
Step 6: Set the equations from Step 4 equal to each other to solve for BC:
BC * tan(30 degrees) = (18 - BC) * tan(58 degrees)
Step 7: Solve the equation from Step 6 to find the value of BC:
BC * tan(30 degrees) = (18 - BC) * tan(58 degrees)
BC * 0.5774 = (18 - BC) * 1.601
0.5774BC = 28.818 - 1.601BC
0.5774BC + 1.601BC = 28.818
2.1784BC = 28.818
BC = 28.818 / 2.1784
BC ≈ 13.23
Step 8: Substitute the value of BC into one of the original equations from Step 4 to find the value of h:
h = BC * tan(30 degrees)
h = 13.23 * 0.5774
h ≈ 7.64
Therefore, the height of the jet above the sea level is approximately 7.64 meters.
To find the height of the jet above the sea level, we can use trigonometry and the concept of angles of depression.
Let's start by drawing a diagram to visualize the information given:
B C
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====================================== jet A
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Here, jet A is flying above the two ships B and C, which are 18 km apart. The angles of depression of B and C from the jet A are given as 60 degrees and 32 degrees, respectively.
Now, let's consider the triangle formed by the line connecting jet A to ship B, the line connecting jet A to ship C, and the line connecting ship B to ship C.
We can use the tangent function to relate the height of the jet A, the distance between jet A and ship B, and the angle of depression from jet A to ship B.
tan(60 degrees) = height of A / distance from A to B
In this case, the height of A is the unknown variable that we want to find. Therefore, we can rearrange the equation to solve for the height:
height of A = distance from A to B * tan(60 degrees)
Similarly, we can use the tangent function with the angle of depression from jet A to ship C:
tan(32 degrees) = height of A / distance from A to C
Again, rearrange the equation to solve for the height:
height of A = distance from A to C * tan(32 degrees)
Since we know that the distance between ship B and ship C is 18 km, we have:
distance from A to B + distance from A to C = 18 km
Substituting the heights calculated from the above equations, we get:
distance from A to B * tan(60 degrees) + distance from A to C * tan(32 degrees) = 18 km
Now, let's substitute the values and calculate the height of the jet:
height of A = distance from A to B * tan(60 degrees)
height of A = 18 km * tan(60 degrees)
height of A = 18 km * 1.732
height of A = 31.176 km (rounded to 3 decimal places)
Therefore, the height of the jet above sea level is approximately 31.176 km, or 31,176 meters (rounded to the nearest meter).