Problem 7: Reasoning
Kent builds a skate ramp with 2 meters of wood. She wants the vertical height of the ramp to be 1 meter. What does the angle of elevation need to be. 
Problem 8: Reasoning
A ship is sighted from the top of a lighthouse.
The angle of depression from the lighthouse to the ship is 45 degrees. The distance from the top of the lighthouse is 4 miles. Calculate the horizontal distance of the ship from the bottom of the lighthouse. Give your answer correct to 2 decimal places.  
Problem 9: Problem -solving 
A rectangular lawn has a diagonal path running across it. The lawn is 10m wide and 15m long. Work out the length of the path. Give your answer in surd form

Let's work through each of these problems one by one.

### Problem 7: Skate Ramp (Angle of Elevation)

We are given:
- The length of the ramp (hypotenuse) is 2 meters.
- The vertical height is 1 meter.

We need to find the angle of elevation, denoted by \( \theta \).

From trigonometry, we know that:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}
\]
To find \( \theta \), we take the inverse sine:
\[
\theta = \sin^{-1}\left(\frac{1}{2}\right)
\]

Let's calculate the angle.

### Problem 8: Lighthouse and Ship (Horizontal Distance)

We are given:
- The angle of depression is \( 45^\circ \).
- The distance from the top of the lighthouse directly to the ship (the hypotenuse of the triangle) is 4 miles.

We need to calculate the horizontal distance of the ship from the base of the lighthouse.

Since the angle of depression is \( 45^\circ \), the horizontal and vertical distances in this right triangle are equal. Thus, the horizontal distance will be:
\[
\text{Horizontal distance} = \text{Hypotenuse} \times \cos(45^\circ)
\]

Let’s calculate that.

### Problem 9: Lawn Path (Length of the Diagonal)

We are given:
- The width of the lawn is 10 meters.
- The length of the lawn is 15 meters.

We need to calculate the length of the diagonal path, which forms the hypotenuse of a right triangle.

Using the Pythagorean theorem:
\[
\text{Length of diagonal} = \sqrt{(10^2 + 15^2)}
\]

Since the problem asks for the answer in surd form, we will keep the square root.

---

Now, I will compute the results for each of these problems.import math

# Problem 7: Skate Ramp (Angle of Elevation)
# Given: height = 1 meter, hypotenuse = 2 meters
angle_elevation = math.degrees(math.asin(1 / 2))

# Problem 8: Lighthouse and Ship (Horizontal Distance)
# Given: angle of depression = 45 degrees, hypotenuse = 4 miles
horizontal_distance = 4 * math.cos(math.radians(45))

# Problem 9: Lawn Path (Length of Diagonal)
# Given: width = 10 meters, length = 15 meters
diagonal_length_surd = math.sqrt(10**2 + 15**2)

angle_elevation, horizontal_distance, diagonal_length_surdHere are the results for each of the problems:

### Problem 7: Skate Ramp (Angle of Elevation)
The angle of elevation needs to be approximately \( 30^\circ \).

### Problem 8: Lighthouse and Ship (Horizontal Distance)
The horizontal distance of the ship from the bottom of the lighthouse is approximately 2.83 miles.

### Problem 9: Lawn Path (Length of Diagonal)
The length of the diagonal path is \( \sqrt{325} \) meters, which is approximately 18.03 meters in decimal form.

---

Do you have any questions or would you like further clarification on any of these problems?

Here are some related questions to think about:
1. How does changing the vertical height affect the angle of elevation in Problem 7?
2. What would happen if the angle of depression in Problem 8 was smaller or larger than \( 45^\circ \)?
3. Can you find the exact diagonal length in meters without converting to decimals in Problem 9?
4. What would the angle of elevation be if the height in Problem 7 was greater than the length of the ramp?
5. How does the Pythagorean theorem apply to solving problems involving right triangles like in Problem 9?

**Tip**: Always remember that trigonometric functions like sine, cosine, and tangent are key to solving right triangle problems involving angles and lengths!

Explore solutions to mathematical problems involving trigonometry and geometry. Learn how to calculate the angle of elevation for a skate ramp, determine the horizontal distance from a lighthouse to a ship using trigonometric functions, and find the diagonal length of a rectangular lawn's path. Suitable for Grades 9-12.

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