A plane is flying at a height of 1000 feet. A passenger on the plane sees a tree and realizes the angle of depression to the top of the tree is 15 degrees and the angle of depression to the bottom of the tree is 25 degrees. How tall is the tree?
What a silly and unrealistic problem,
anyway ....
I made a sketch
the main working triangle has the height of the tree, its opposite angle as 10°, and the other angles as 105° and 65°
The side opposite the 105° is also the hypotenuse of a right-angled triangle with vertical side 1000 and base angle 25° (alternate angle to your angle of depression)
let that hypotenuse be h
sin25 = 1000/h
h = 1000/sin25 = 2366.2
now in the other triangle, by the sine law
tree/sin10 = 2366.2/sin105
tree = 2366.2sin10/sin105 = 425.4 ft
wow !
Since the tallest trees in the world, the sequoias of the west coast all are less than 400 ft ..........
To find the height of the tree, we can use trigonometry and the concept of angle of depression. Let's break down the problem step by step.
Step 1: Understand the situation
We have a plane flying at a height of 1000 feet. From the plane, a passenger observes the top and bottom of a tree, measuring the angles of depression to be 15 degrees and 25 degrees respectively. We need to find the height of the tree from the ground.
Step 2: Visualize the problem
To make it easier to understand, let's draw a diagram. Sketch a right-angled triangle representing the situation. Label the plane's height as 1000 feet, the tree's height as h (what we are trying to find), the distance from the plane to the tree as d, and the angles of depression as 15 degrees and 25 degrees.
|\
| \ h
| \
| \
|___\
--->d<---
Step 3: Identify relevant trigonometric functions
Since we are dealing with angles of depression, we can use the tangents of these angles to solve for the tree's height. The tangent of an angle can be defined as the ratio of the opposite side (h) to the adjacent side (d).
Step 4: Set up equations
Using the definition of the tangent function, we can set up two equations:
For the top of the tree: tan(15 degrees) = h / d
For the bottom of the tree: tan(25 degrees) = (h + 1000) / d
Step 5: Solve the equations
Now we can solve these two equations to find the value of h, the height of the tree.
First, rearrange the equation for the top of the tree to solve for h: h = d * tan(15 degrees)
Next, substitute this value of h into the equation for the bottom of the tree: tan(25 degrees) = (d * tan(15 degrees) + 1000) / d
Simplify the equation and isolate d: d = 1000 / (tan(25 degrees) - tan(15 degrees))
Now, put this value of d back into the equation for the top of the tree to find h: h = d * tan(15 degrees)
Finally, calculate the height of the tree by adding h to the initial height of the plane: height of the tree = h + 1000 feet
By plugging in the values and solving the equations, we can obtain the actual height of the tree.