The angles of elevation to an airplane from
two points A and B on level ground are 50◦
and 67◦
, respectively. The points A and B
are 3.9 miles apart, and the airplane is east of
both points in the same vertical plane.
Find the altitude of the plane.
Answer in units of miles. Show all work.
Make a sketch.
I used your A and B as points on the ground, P as the position of the plane and R as a point on the ground directly below P. I let angle A = 50°
Enter your given data.
angle ABP = 113°
Therefore angle APB = 17°
by the sine law:
PB/sin50 = 3.9/sin17
PB = 3.9sin50/sin17
= ...
In the right-angled triangle BRP
sin 67 = PR/BP
PR = BPsin 67
= 3.9 sin50 sin67/sin17
= appr 9.4
or (this is the more traditional way of doing this)
Let BR = x and PR = h
In triangle ARP:
tan 50 = h/(x+3.9)
h = (x+3.9)tan50
in triangel BRP:
tan 67 = h/x
h = xtan67
xtan67 = (x+3.9)tan50 = xtan50 + 3.9tan50
xtan67 -xtan50 = 3.9tan50
x(tan67-tan50) = 3.9tan50
x = 3.9tan50/(tan67-tan50)
back to h = xtan67
h = 3.9tan50 tan67/(tan67-tan50)
= appr 9.4 , just like above
To find the altitude of the plane, we can use trigonometry. Let's denote the altitude as "h" and the distance from Point A to the plane as "x".
First, let's consider the right-angled triangle formed by the plane, Point A, and the vertical line from the plane's altitude. The angle of elevation from Point A is 50 degrees. Using trigonometry, we can write:
tan(50 degrees) = h / x
Next, let's consider another right-angled triangle formed by the plane, Point B, and the vertical line from the plane's altitude. The angle of elevation from Point B is 67 degrees. Using trigonometry again, we can write:
tan(67 degrees) = h / (x + 3.9)
Now we have two equations with two unknowns (h and x):
Equation 1: tan(50 degrees) = h / x
Equation 2: tan(67 degrees) = h / (x + 3.9)
To solve this system of equations, we can use substitution or elimination. Let's use substitution. Rearrange Equation 1 to solve for h:
h = x * tan(50 degrees)
Now substitute this expression for h in Equation 2:
tan(67 degrees) = (x * tan(50 degrees)) / (x + 3.9)
To solve for x, we can simplify the equation by multiplying both sides by (x + 3.9):
(x + 3.9) * tan(67 degrees) = x * tan(50 degrees)
Expanding both sides:
x * tan(67 degrees) + 3.9 * tan(67 degrees) = x * tan(50 degrees)
Rearranging the terms:
x * (tan(67 degrees) - tan(50 degrees)) = 3.9 * tan(67 degrees)
Divide both sides by (tan(67 degrees) - tan(50 degrees)):
x = (3.9 * tan(67 degrees)) / (tan(67 degrees) - tan(50 degrees))
Now we can substitute this value of x back into Equation 1 to find h:
h = x * tan(50 degrees)
Finally, substitute the values of tan(50 degrees) and tan(67 degrees) from a calculator, and plug them into the equations above to find the values of h and x. The altitude of the plane is the value we found for h, and it will be in units of miles.