The length of a shadow of a tree is 130 feet when the angle of elevation of the sun is θ°.
a) Write the height h of the tree as a function of θ.
b) θ=10° What's height?
θ=15° Height?
θ=20° How tall is the height?
θ=25° What is height?
a) The height h of the tree can be written as a function of θ using trigonometry. We can use the tangent function because it relates the angle of elevation to the height and length of the shadow.
Using the tangent function:
tan(θ) = h / 130
Simplifying the equation, we can express the height as:
h = 130 * tan(θ)
b) Let's substitute the given values of θ and calculate the height:
For θ = 10°:
h = 130 * tan(10°) ≈ 23.24 feet
For θ = 15°:
h = 130 * tan(15°) ≈ 35.94 feet
For θ = 20°:
h = 130 * tan(20°) ≈ 49.27 feet
For θ = 25°:
h = 130 * tan(25°) ≈ 65.97 feet
So, the height of the tree at different angles of elevation is approximately:
- At θ = 10°, the height is 23.24 feet.
- At θ = 15°, the height is 35.94 feet.
- At θ = 20°, the height is 49.27 feet.
- At θ = 25°, the height is 65.97 feet.
a) The height of the tree, h, can be determined using trigonometry. We can use the tangent function, which relates the opposite side (the height of the tree) to the adjacent side (the length of the shadow). The equation would be:
tan(θ) = h / 130
Rearranging the equation to solve for h, we have:
h = 130 * tan(θ)
b) Given:
- θ = 10°
- θ = 15°
- θ = 20°
- θ = 25°
Using the equation from part a, we can substitute these values to find the height:
For θ = 10°:
h = 130 * tan(10°)
For θ = 15°:
h = 130 * tan(15°)
For θ = 20°:
h = 130 * tan(20°)
For θ = 25°:
h = 130 * tan(25°)
To answer this question, we need to use trigonometry. We can use the concept of similar triangles to determine the height of the tree based on the given angle of elevation and the length of the shadow.
Let's start by drawing a diagram to visualize the situation. The tree, the length of its shadow, and the angle of elevation are as follows:
```
Tree
|
|\
| \
Height | \ Shadow
(h) | \
|θ \
```
a) Write the height h of the tree as a function of θ:
To find the height h as a function of θ, we need to consider the relationship between the height, the shadow length, and the angle of elevation. We know that the tangent of an angle is equal to the opposite side divided by the adjacent side in a right triangle. In this case, the opposite side is the height h, and the adjacent side is the length of the shadow.
So, we can write:
tan(θ) = h / Shadow length
Rearranging the equation, we have:
h = Shadow length * tan(θ)
Therefore, the height h of the tree is given by the equation h = Shadow length * tan(θ).
b) Now, let's calculate the height for each given angle:
For θ = 10°:
h = 130 * tan(10°)
Using a calculator, we find h ≈ 23.01 feet.
For θ = 15°:
h = 130 * tan(15°)
Using a calculator, we find h ≈ 34.26 feet.
For θ = 20°:
h = 130 * tan(20°)
Using a calculator, we find h ≈ 47.16 feet.
For θ = 25°:
h = 130 * tan(25°)
Using a calculator, we find h ≈ 63.38 feet.
So, the height of the tree for each given angle is approximately:
θ = 10°: 23.01 feet
θ = 15°: 34.26 feet
θ = 20°: 47.16 feet
θ = 25°: 63.38 feet.
a. h = 130*tan(theta).
b. h = 130*tan10 = 22.9 Ft.