An airplane is flying from the west directly towards two forest fires that are east of the airplane. From the airplane, the angle of depression to one fire is 43 (degrees) and 20 (degrees) to the other fire. The airplane is flying at an altitude of 2500ft. what is the distance between of the two fires and the nearest foot?

Question

An airplane is flying from the west directly towards two forest fires that are east of the airplane. From the airplane, the angle of depression to one fire is 43 (degrees) and 20 (degrees) to the other fire. The airplane is flying at an altitude of 2500ft. what is the distance between of the two fires and the nearest foot?

Answer 1

I really need help ASAP!!!! please anyone!!!

Answer 2

distance to closest fire ... d / 2500 ft = tan(90º - 43º)

... d = 2500 ft * tan(47º)

distance to farthest fire ... d / 2500 ft = tan(90º - 20º)
... d = 2500 ft * tan(70º)

Answer 3

To find the distance between the two fires, we can use trigonometry and the concept of angles of depression.

Let's consider the triangle formed by the airplane, one fire, and the line connecting the airplane to the fire. We can label this triangle as Triangle A.

In Triangle A, the angle of depression from the airplane to one fire is 43 degrees. This means that the angle between the horizontal line and the line connecting the airplane to the fire is also 43 degrees.

Now, let's consider the other triangle formed by the airplane, the other fire, and the line connecting the airplane to the other fire. We can label this triangle as Triangle B.

In Triangle B, the angle of depression from the airplane to the other fire is 20 degrees. This means that the angle between the horizontal line and the line connecting the airplane to the other fire is also 20 degrees.

Since the angles between the horizontal and the two lines connecting the airplane to the fires are equal, we can conclude that the two triangles (Triangle A and Triangle B) are similar triangles.

Now, to find the distance between the two fires, we need to find the lengths of the sides of Triangle A and Triangle B. Let's label the distance between the airplane and the first fire as x, and the distance between the airplane and the other fire as y.

Since both triangles are similar, we can set up the following proportion:

x / y = tan(43) / tan(20)

Using this proportion, we can find the value of x and y.

Let's calculate it:

Using a scientific calculator to find the tangent of the angles:

tan(43) ≈ 0.9325
tan(20) ≈ 0.3640

Now, let's substitute these values into the proportion:

x / y = 0.9325 / 0.3640

To isolate x, we can cross-multiply:

x = (y * 0.9325) / 0.3640

Now, we need to find the value of y. We can use the altitude of the airplane and the angle of depression to the second fire to find this.

In Triangle B, the opposite side is the altitude of the airplane, which is 2500 ft. The adjacent side is y.

Using the tangent function, we can write:

tan(20) = opposite / adjacent

tan(20) = 2500 / y

Now, let's solve for y:

y = 2500 / tan(20)

Using a scientific calculator:

y ≈ 2500 / 0.3640

y ≈ 6868.13 ft

Now that we have the value of y, we can substitute it back into the equation for x:

x = (6868.13 * 0.9325) / 0.3640

Using a calculator:

x ≈ 17733.72 ft

Therefore, the distance between the two fires is approximately 17733.72 feet.