an observer at A looks due north and sees a meteor with an angle of elevation of 70 degrees. At the same instant, another observer 30 miles east of A seed the same meteor and approximates its position as N 50 degrees W but fails to note its angle of elevation. Find the height of meteor and its distance from A.
I don't know how good you are at drawing a 3D diagram.
Start with a NSEW grid and place A at the origin.
(I angled my North and East axes at about 60° to each other, but marked the angle at A as ∟ (90°)
Draw in MC, the meteor, where C is on the North axis.
We want MC, a vertical side in the rightangled triangle MAC , where C is the 90° angle and angle A = 70°
Mark position B on the East axis, so that AB = 30 miles
Joint BC and BM to have another right-angled triangle BMC, with angle C = 90°
From B, sketch another NORTH axis BD, so that BD || AC
We are told that angle DBC = 50° (N 50 W)
so in the right-angled triangle ABC, angle ABC = 90-50 = 40°
Now for our calculations:
In triangle ABC,
AC/AB = tan 40°
AC = ABtan40
AC= 30tan40°
in triangle MAC,
MC/AC = tan70
MC = ACtan70
= 30tan40tan70
= appr 69.2 miles
for the distance MA,
sin 70 = MC/MA
MA = MC/sin70 =73.6 miles
check my arithmetic
To find the height of the meteor and its distance from point A, we can use trigonometry and the information provided.
Let's first label the given information:
Observer A:
- Angle of elevation: 70 degrees
Observer B:
- Approximated position: N 50 degrees W (This means the angle between B-A direction and the north direction is 50 degrees.)
Now, let's solve for the height of the meteor:
Step 1: Find the angle of elevation at observer B.
To find the angle of elevation at observer B, we can use the knowledge that the angle between B-A and the north direction is 50 degrees. Since the meteor's path is due north, the angle of elevation at B is complementary to the angle between B-A and the north direction.
Angle of elevation at B = 90 degrees - 50 degrees = 40 degrees
Step 2: Determine the distance from observer B to the meteor.
To determine the distance from observer B to the meteor, we can use trigonometry and the fact that both observer A and observer B see the same meteor.
Let's assume that the distance from observer A to the meteor is x miles, and the height of the meteor is h miles.
Using observer A's information:
tan(70 degrees) = h / x
Using observer B's information:
tan(40 degrees) = h / (x + 30) (since observer B is 30 miles east of observer A)
Step 3: Solve the equations simultaneously.
By solving the two equations obtained in Step 2 simultaneously, we can find the values of x and h.
First, rearrange the equations:
tan(70 degrees) = h / x --> Equation 1
tan(40 degrees) = h / (x + 30) --> Equation 2
Divide Equation 1 by Equation 2:
[tan(70 degrees) / tan(40 degrees)] = (h / x) / (h / (x + 30))
Simplify:
tan(70 degrees) * (x + 30) = tan(40 degrees) * x
Now, plug in the values of the tangent of the angles (you can use a calculator):
1.537 * (x + 30) = 0.839 * x
Solve this equation to find the value of x, which represents the distance from observer A to the meteor.
Then, substitute the value of x back into one of the equations to find the height of the meteor (h).
Once you have solved these equations, you will have the height of the meteor and its distance from observer A.