. From the top of the 800-foot-tall Cartalk Tower, Tom sees a plane; the angle of elevation is 67°. At the

same instant, Ray, who is on the ground 1 mile from the building, notes that his angle of elevation to the
plane is 81° and that his angle of elevation to the top of Cartalk Tower is 8.6°. Assume Tom, Ray, and the
airplane are in a plane perpendicular to the ground. How high is the airplane?

I used right triangles and it gave me 9039.846ft but my teacher said that it was incorrect.

To find the height of the airplane, we can use trigonometry and the concept of similar triangles.

Let's break down the problem and label the given information:

- Tom is on top of the Cartalk Tower (800 ft tall) and observes the airplane with an angle of elevation of 67°.
- Ray is on the ground, 1 mile (5280 ft) away from the tower. He observes the airplane with an angle of elevation of 81° and the top of Cartalk Tower with an angle of elevation of 8.6°.

Now, let's use these angles to find the height of the airplane.

First, let's focus on Ray's observation. We can create a right triangle with Ray at the base, the airplane at the top, and the line of sight as the hypotenuse.

Using trigonometric ratios, we can say that:

tan(81°) = height of the airplane / distance from Ray to the airplane

Therefore, height of the airplane = distance from Ray to the airplane * tan(81°)

Since Ray is 1 mile (5280 ft) away from the tower, we have:

height of the airplane = 5280 ft * tan(81°) ≈ 24172.586 ft

So, according to Ray's observation, the height of the airplane is approximately 24172.586 ft.

Now, let's consider Tom's observation. We can create a right triangle with Tom at the base, the airplane at the top, and the line of sight as the hypotenuse.

Using trigonometric ratios, we can say that:

tan(67°) = height of the airplane / height of Cartalk Tower

Therefore, height of the airplane = height of Cartalk Tower * tan(67°)

Since the Cartalk Tower is 800 ft tall, we have:

height of the airplane = 800 ft * tan(67°) ≈ 2468.196 ft

According to Tom's observation, the height of the airplane is approximately 2468.196 ft.

Now, we have two different values for the height of the airplane: 24172.586 ft from Ray's observation and 2468.196 ft from Tom's observation. Since these values are significantly different, it seems there may be an error in the calculations or assumptions made.

I recommend rechecking the values and calculations to ensure accuracy or consulting with your teacher for further clarification.

To find the height of the airplane, we can use the concept of trigonometry and create a diagram.

Let's label the height of the airplane as 'x'.

Based on the information given, we have two right triangles: one formed by Tom's line of sight and the height of the tower, and another formed by Ray's line of sight to the tower and the airplane.

In triangle 1 (Tom's triangle):

The height of the tower is 800 feet.

The angle of elevation from Tom to the airplane is 67°.

In triangle 2 (Ray's triangle):

The distance from Ray to the tower is 1 mile, which is 5280 feet.

The angle of elevation from Ray to the airplane is 81°.

The angle of elevation from Ray to the top of the tower is 8.6°.

To solve for 'x', we will use the tangent function in triangle 1:

tan(67°) = x / 800

Solving for 'x':

x = 800 * tan(67°)
x ≈ 2140.79 feet (rounded to two decimal places)

Therefore, the height of the airplane is approximately 2140.79 feet. This differs from your calculation of 9039.846 feet, which may have been due to an error or incorrect substitution of values.

Let

x = distance from Ray to the point under the plane
h = height of plane

You can see from the diagram that

(h-800)/(5280+x) = tan67°
h/x = tan81°

Now just eliminate x and solve for h. I get 21,119
(c-800)/tan(67°) -5280 = c/tan(81°)