
34. <$• MULTIPLE REPRESENTATIONS You are climbing Mount Massive in Colorado. You are at an altitude of 11,200 feet. You measure the angle of elevation to a ridge above you to be 29.4°. The distance (along the face of the mountain) between you and the ridge is 6315 feet.
a. Drawing a Diagram Draw a diagram that represents this situation.
b. Writing an Equation Write and solve an equation to find the altitude of the ridge.

Let's break this down step by step.

### Part (a) - Drawing a Diagram

You are standing at an altitude of 11,200 feet, and you measure the angle of elevation to a ridge above you to be 29.4°. The distance between you and the ridge, measured along the slope of the mountain, is 6,315 feet.

In this scenario, the situation forms a right triangle:
- The distance along the mountain (hypotenuse) is 6,315 feet.
- The angle of elevation is 29.4°.
- The height difference between your altitude and the ridge is the opposite side of the triangle.

You can visualize this as a right triangle where:
- The **base** is horizontal (the distance between the two positions, but we don't need to find it).
- The **height** is the difference in altitudes (which we are looking for).
- The **hypotenuse** is the line between your current position and the ridge, which is 6,315 feet.
  
We’ll use trigonometry to calculate the height difference.

### Part (b) - Writing and Solving an Equation

We can use the sine function because we have the hypotenuse and the angle, and we are solving for the opposite side, which is the height difference.

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

Here:
- $\theta = 29.4^\circ$ (angle of elevation),
- Hypotenuse = 6315 feet (distance along the mountain),
- Opposite = the vertical height difference between you and the ridge (which we need to find).

So, the equation becomes:

\[
\sin(29.4^\circ) = \frac{\text{height difference}}{6315}
\]

Now, solving for the height difference:

\[
\text{height difference} = 6315 \times \sin(29.4^\circ)
\]

Let me calculate that:

\[
\text{height difference} \approx 6315 \times 0.4916
\]
\[
\text{height difference} \approx 3103.3 \text{ feet}
\]

### Final Step: Finding the Altitude of the Ridge

To find the total altitude of the ridge, add the height difference to your current altitude (11,200 feet):

\[
\text{Altitude of the Ridge} = 11,200 + 3103.3 \approx 14,303.3 \text{ feet}
\]

So, the altitude of the ridge is approximately **14,303 feet**.

---

Would you like further clarification on any of the steps or concepts? Here are some related questions you might find interesting:

1. How can you use other trigonometric functions (cosine, tangent) in similar problems?
2. How would the equation change if you were measuring a depression angle instead of an elevation?
3. What if the distance along the face of the mountain were different, say 7000 feet? How would that impact the altitude of the ridge?
4. What are other practical uses of trigonometry in outdoor activities like hiking or climbing?
5. How could this method help you estimate the altitude of different mountain peaks?

**Tip:** When solving problems involving right triangles, it’s useful to sketch a diagram to clearly identify the sides relative to the angle you’re working with!

You are climbing Mount Massive in Colorado. You are at an altitude of 11,200 feet. You measure the angle of elevation to a ridge above you to be 29.4°. The distance (along the face of the mountain) between you and the ridge is 6315 feet. a. Draw a diagram that represents this situation. b. Write and solve an equation to find the altitude of the ridge.

Solve a real-world trigonometry problem involving Mount Massive in Colorado. Calculate the altitude of a ridge using the sine function and the angle of elevation. The problem explores the use of right triangles and trigonometric ratios. Suitable for students in Grades 9-11.

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