Two observers who are 5 km apart simultaneously sight a small airplane flying between them. One observer measures a 51.0-degree inclination angle, while the other observer measures a 40.5-degree inclination angle. At what altitude is the
airplane flying?
To find the altitude of the airplane, we can use the concept of trigonometry. Let's denote the distance between the two observers as 'd' and the altitude of the airplane as 'h'.
Step 1: Draw a diagram to visualize the problem. Label the two observers as Observer A and Observer B, and the distance between them as 5 km.
Step 2: From Observer A, draw a line representing the line of sight to the airplane. This line makes an angle of 51.0 degrees with the ground.
Step 3: From Observer B, draw a line representing the line of sight to the airplane. This line makes an angle of 40.5 degrees with the ground.
Step 4: Draw another line connecting the airplane to the ground, forming a triangle between the airplane, Observer A, and Observer B.
Step 5: Notice that this triangle is a right triangle, with the distance between Observer A and Observer B as the base, and the altitude of the airplane as the height.
Step 6: Now, we can use the tangent function to find the altitude. The tangent of an angle is equal to the opposite side divided by the adjacent side.
For Observer A:
tan(51.0 degrees) = h/d
For Observer B:
tan(40.5 degrees) = h/d
Step 7: We can rearrange the equations to solve for the altitude, 'h':
h = d * tan(51.0 degrees)
h = d * tan(40.5 degrees)
Step 8: Substitute the value of 'd' (which is 5 km) into the equations:
h = 5 km * tan(51.0 degrees)
h = 5 km * tan(40.5 degrees)
Step 9: Use a calculator to find the values of the tangents and multiply them by 5 km to calculate the altitude.
h ≈ 5 km * tan(51.0 degrees) ≈ 5 km * 1.285 ≈ 6.425 km
h ≈ 5 km * tan(40.5 degrees) ≈ 5 km * 0.862 ≈ 4.310 km
Therefore, based on the measurements provided, the altitude of the airplane is approximately 6.425 km as observed from Observer A and 4.310 km as observed from Observer B.
To find the altitude at which the airplane is flying, we can use trigonometry and the concept of similar triangles. Here's how you can solve this problem step by step:
Step 1: Draw a diagram
Start by drawing a diagram with the two observers (A and B) that are 5 km apart and the airplane (C) flying between them. Label the distance between the observers as "d" (which is 5 km), the angle of inclination measured by observer A as "angle A" (which is 51.0 degrees), the angle of inclination measured by observer B as "angle B" (which is 40.5 degrees), and the altitude of the airplane as "h" (which is what we need to find).
Step 2: Identify similar triangles
Since the observers are at a fixed distance from each other and both are sighting the airplane, the triangles formed by the observers and the airplane are similar. In similar triangles, corresponding angles are equal, and the ratios of corresponding sides are equal.
Step 3: Set up the trigonometric equation
Using the concept of similar triangles, we can set up the following equation based on the tangent of the angles:
tan(angle A) = h / x (where x is the horizontal distance from observer A to the airplane)
tan(angle B) = h / (d - x) (where d is the distance between the observers and x is the horizontal distance from observer A to the airplane)
Step 4: Solve the equations
Rearrange both equations to solve for x:
x = h / tan(angle A)
d - x = h / tan(angle B)
Rewrite the second equation as:
d = h / tan(angle B) + x
Substitute the value of x from the first equation into the second equation:
d = h / tan(angle B) + (h / tan(angle A))
Simplify the equation:
d = h * (1 / tan(angle B) + 1 / tan(angle A))
Multiply both sides by tan(angle B) * tan(angle A):
d * tan(angle B) * tan(angle A) = h * (tan(angle A) + tan(angle B))
Divide both sides by (tan(angle A) + tan(angle B)):
h = (d * tan(angle B) * tan(angle A)) / (tan(angle A) + tan(angle B))
Step 5: Calculate the altitude
Now, substitute the given values into the equation and solve for h:
Using the values given in the question, d = 5 km, angle A = 51.0 degrees, and angle B = 40.5 degrees:
h = (5 km * tan(40.5 degrees) * tan(51.0 degrees)) / (tan(51.0 degrees) + tan(40.5 degrees))
Using a calculator, evaluate the expression to get the value of h.
Therefore, the altitude at which the airplane is flying can be determined using the above steps.
Well, it seems like the airplane is caught in an awkward situation where it's being watched from different angles. The first observer is like, "Hey airplane, I see you at a 51-degree angle!" And the second observer is like, "Wait a minute, I see you at a 40.5-degree angle!"
Now, let's help this confused airplane out. We can draw a diagram where the observers are at the vertices of a triangle, and the airplane is somewhere in the middle. The distance between the observers is 5 km, which makes this triangle one big geometry mess.
To find the altitude of the airplane, we need some trigonometry magic. We can use the tangent function, which is like a wizard's wand in these situations. The tangent of the inclination angle is equal to the altitude divided by the distance between the observers.
So, we have:
tan(51 degrees) = altitude / 5 km
tan(40.5 degrees) = altitude / 5 km
Now, let's solve these equations to find the altitude.
Just give me a second to compute the results... *beep boop beep*
Okay, according to my calculations, the altitude of the airplane is approximately [insert answer here]. Keep in mind, though, that flying at this altitude might cause some neck cramps from constantly adjusting to different observers' angles.
Draw a triangle ABC , A and B the position of the observers and C the position of the plane.
Drop an altitude from C to AB, call it CD.
Label angle A as 51 and angle B as 40.5 degrees
Label AD as x, then DB = 5-x
In triangle ACD, cot51 = x/h
x = hcot51
In triangle DBC cot 40.5 = (5-x)/h
5-x = hcot40.5 or
x = 5 - hcot40.5
then hcot51 = 5-hcot40.5
hcot51 + hcot40.5 = 5
h = 5/(cot51 + cot40.5)
= .....