Find the height of a tree if the angle of elevation of its top changes
from 20°. to 40° as the observer advances 75 ft toward its base.
Provide an Illustration.
Oh, I see! So you want to know the height of the tree based on the change in the angle of elevation as the observer moves closer. Well, let me put on my Clown Bot thinking cap and try to give you an answer that's as amusing as it is informative!
Imagine the observer is strolling towards the tree, feeling all adventurous and eager to solve the mystery of its height. As they walk 75ft closer to the tree, they're basically pulling a "slow and steady" move on the tree, like a sneaky game of hide-and-seek.
Now, let's focus on the change in the observer's perspective. At the starting point, the angle of elevation to the top of the tree is at 20°. But as they move forward, inch by inch, trying not to step on any pesky squirrels, the angle of elevation increases to a mighty 40°. Ah, the dramatic twist in this grand tree height-seeking adventure!
So, using this new information, we can set up our calculation. We have our trusty right-angled triangle, with the vertical height of the tree as the unknown, the 75ft the observer advanced as the horizontal distance, and the two angles of elevation at our disposal.
To figure out the height, we can use some fancy math called trigonometry! Specifically, we'll use the tangent function.
tan(20°) = height / distance (before advancing) - Equation A
tan(40°) = height / (distance before advancing + 75ft) - Equation B
Now, here comes the "ta-da!" moment. We have two equations with two unknowns! We can solve this system of equations to find the height of that wily tree.
But, alas, illustrating this calculation might be a bit out of my comedic range. So, picture this: Picture a clown (me!) juggling triangles and equations while trying not to drop anything valuable (like your understanding of this math problem). And in the end, voila! The height of the tree is revealed!
Now, it's time for you to take the stage, solve these equations, and uncover the majestic height of that tree!
To find the height of the tree, we can use trigonometry and set up a right triangle. Let's refer to the height of the tree as "h" and the distance from the observer to the base of the tree as "x".
Step 1: Draw a diagram to represent the situation. Label the height of the tree as "h", the distance from the observer to the base as "x", and the angle of elevation as 20° and 40° respectively.
|
|\
| \
| \
| \ h
| \
x |-----\
| 20° \
| \
| \
----------------------------------------------------------------------------------
Step 2: Use the tangent function to set up an equation. Tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
Let's use 20° angle of elevation: tan(20) = h / x
Similarly, for 40° angle of elevation: tan(40) = h / (x - 75)
Step 3: Solve the equation for h.
From the first equation, we can express h in terms of x:
h = x * tan(20)
Substitute this value of h in the second equation:
tan(40) = (x * tan(20)) / (x - 75)
Step 4: Solve for x.
Multiply both sides of the equation by (x - 75) to get rid of the denominator:
(x - 75) * tan(40) = x * tan(20)
Expand the equation:
x * tan(40) - 75 * tan(40) = x * tan(20)
Rearrange the equation:
x * (tan(40) - tan(20)) = 75 * tan(40)
Divide both sides by (tan(40) - tan(20)):
x = (75 * tan(40)) / (tan(40) - tan(20))
Step 5: Substitute the value of x into either equation to find h.
Using the equation h = x * tan(20), substitute the value of x we just found:
h = [(75 * tan(40)) / (tan(40) - tan(20))] * tan(20)
Step 6: Calculate h.
Calculate the value of h using a calculator:
h ≈ 101.32 ft
Therefore, the approximate height of the tree is 101.32 feet.
Note: The illustration cannot be provided in this text format.
To find the height of the tree, we can use trigonometry and set up a right triangle between the observer, the base of the tree, and the top of the tree. Here's how to solve this problem step by step:
1. Let's denote the height of the tree as h. The initial angle of elevation from the observer to the top of the tree is 20°, and after advancing 75 ft toward the base of the tree, the angle of elevation becomes 40°.
2. Draw a diagram representing the situation. Label the bottom of the diagram as "Observer," the top of the triangle as "Tree Top," and the bottom of the triangle as "Tree Base." Indicate the angle of elevation of 20° and 40° accordingly. Also, mark the distance of 75 ft from the "Observer" to the "Tree Base."
3. Now, we need to break down the problem into two separate right triangles. The first triangle consists of the "Observer," "Tree Base," and a right angle. The second triangle includes the "Observer," "Tree Top," and the same right angle.
4. Calculate the base of the first triangle (distance between the observer and the base of the tree) using the given information. The distance is given as 75 ft.
5. We can use the tangent function to find the height of the tree in the first triangle. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the height of the tree (h) is the opposite side, and the base of the first triangle (75 ft) is the adjacent side. So, we can write the equation as tan(20°) = h/75.
6. Rearrange the equation to solve for h: h = 75 * tan(20°).
7. Use a calculator to find the value of tan(20°) and multiply it by 75. The result will give you the height of the tree.
Illustration:
```
|
|\
h | \
| \
| \ 40°
| \
| \
| \
_____________|_______\_
75 ft
```
Remember to substitute the value of tan(20°) into the equation to get the accurate result for the height of the tree.