an airplane is seen from an observation point on the ground at an angle of elevation of 37 degrees. it is seen from a second observation point on the ground 800 yards away from the first observation point at an angle of elevation of 65 degrees. how far is the airplane from the first observation point?
Ah, the classic case of two observation points and a flying plane! Let's do some math and add a touch of humor along the way.
First, we can start by finding the height of the plane above the ground at the first observation point. We'll call it 'h'.
To find this, we can use the tangent function, because everyone knows that trigonometry loves tangents more than a delicious pastry loves icing. So, using the first observation point, we can say:
tan(37 degrees) = h / x
Where 'x' is the distance between the first observation point and the plane. And let's not forget that 'rad' means "Really Awesome Degree"!
Simplifying this equation, we have:
h = x * tan(37 degrees)
Now that we know the height of the plane above the ground at the first observation point, let's move on to the second observation point and find the distance between the plane and the first observation point.
Again, utilizing the tangent function, we can say:
tan(65 degrees) = h / (x + 800)
Because now we have a horizontal distance of 'x + 800' between the second observation point and the plane.
Simplifying this equation, we get:
h = (x + 800) * tan(65 degrees)
Now, since both 'h' values are equal, we can set them equal to each other and solve for 'x':
x * tan(37 degrees) = (x + 800) * tan(65 degrees)
And now, we unleash the power of algebra to solve for 'x':
x = (800 * tan(65 degrees)) / (tan(37 degrees) - tan(65 degrees))
Remember, always be x-tra careful with your calculations!
Plugging the values into your favorite calculator, you'll find that x is approximately 1024.61 yards.
So, the airplane is about 1024.61 yards away from the first observation point. Hope this answer flies high for you!
To solve this problem, we can use trigonometry. Let's denote the distance from the first observation point to the airplane as "x".
From the first observation point, we have an angle of elevation of 37 degrees. This means that we have a right triangle where the opposite side is x and the adjacent side is the horizontal distance (800 yards).
Using the tangent of 37 degrees, we can write the equation:
tan(37) = opposite/adjacent = x/800
From the second observation point, we have an angle of elevation of 65 degrees. This forms another right triangle where the opposite side is also x and the adjacent side is the horizontal distance (800 yards + x).
Using the tangent of 65 degrees, we can write the equation:
tan(65) = opposite/adjacent = x/(800 + x)
Now we can solve for x by setting these two equations equal to each other and solving for x.
tan(37) = x/800
tan(65) = x/(800 + x)
Solving these two equations simultaneously will give us the value of x, which represents the distance of the airplane from the first observation point.
To find the distance of the airplane from the first observation point, we can use trigonometric functions and the given angles of elevation.
Let's denote the distance between the first observation point and the airplane as "x".
From the first observation point, we have a right triangle. The angle of elevation of 37 degrees is opposite the side "x" and adjacent to the height or altitude of the airplane. Therefore, we can use the tangent function:
tan(37) = (height of the airplane) / x
Next, we consider the second observation point. Now we have another right triangle. The angle of elevation of 65 degrees is opposite the side "x + 800" (since the airplane is at a distance of 800 yards from the first observation point), and adjacent to the same height or altitude of the airplane. Again, we can use the tangent function:
tan(65) = (height of the airplane) / (x + 800)
Now we have two equations with the same height of the airplane. We can set them equal to each other and solve for x:
tan(37) = tan(65) * (x + 800) / x
By solving this equation, we can find the value of x, which represents the distance of the airplane from the first observation point.
Make a sketch
Label the position of the plane as P
label the point directly below P as Q
Label the first observation point A and the second observarion point as B
angle A = 37°
angle PBQ = 65°
AB = 800
You can easily see that angle APB= 28°
In triangle ABP you can find side ATP using the sine law.