A lighthouse keeper in his observation tower 120 feet above the water sees a pontoon airplane and a small fishing boat. If the plane is directly above the boat, how far above sea level is the plane flying, rounded to the nearest foot? Be sure to show your work

To determine the height at which the plane is flying above sea level, we need to consider the distance between the observer and the plane, and the observer's height above the water.

Let's call the height at which the plane is flying "x" feet. Since the observer is 120 feet above the water, the total height from the observer to the plane is 120 + x feet.

Considering the triangle formed by the observer, the plane, and the boat:

Using the Pythagorean theorem, we can determine the distance between the observer and the plane (hypotenuse of the triangle):

(120 + x)^2 = 120^2 + 120^2

(120 + x)^2 = 14400 + 14400

(120 + x)^2 = 28800

Taking the square root of both sides:

120 + x = √28800

120 + x = 169.706

x = 169.706 - 120

x = 49.706

Therefore, the plane is flying approximately 49.706 feet above sea level, rounded to the nearest foot.

To find the height at which the plane is flying above sea level, we can use the concept of similar triangles. Let's consider the observation tower, the plane, and the boat as points forming the vertices of a triangle.

The height of the observation tower, 120 feet, can be considered the vertical side of the triangle (tower height). Since the plane is directly above the boat, the line connecting the top of the tower to the plane will be parallel to the ground (horizontal).

We can consider the height at which the plane is flying above sea level as the other vertical side of the triangle (plane height). Now, we need to find the length of this side.

Since the observation tower and the plane are parallel to each other, we can use the concept of similar triangles. The ratio of the corresponding sides of similar triangles is equal. So, we can set up the following proportion:

Height of tower / Distance from tower to plane = Plane height / Distance from tower to sea level

Substituting the given values, we get:

(120 feet) / Distance from tower to plane = Plane height / Distance from tower to sea level

We know that the distance from the tower to the plane is the same as the distance from the tower to sea level (since the plane is directly above the boat). So, we can rewrite the equation as:

120 feet / Distance from tower to plane = Plane height / Distance from tower to plane

To find the distance from the tower to the plane, we need an additional piece of information. If we have the angle at which the lighthouse keeper is observing the plane, we could use trigonometry (specifically tangent function) to solve for the distance from the tower to the plane. Could you provide the angle at which the lighthouse keeper is observing the plane?

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