Solving for the Indicated Value

Oct 13, 2025 by windailys.com 58 views

Problem 12: Finding the Distance to Dillon's Cabin

Situation

While standing at the top of a mountain, Dillon spots his cabin below. If the vertical height of the mountain is 2400 feet and the angle of depression is 46 degrees, how far is Dillon's cabin from the base of the mountain?

Drawing a Picture

To solve this problem, we need to draw a picture that represents the situation. Let's draw a right triangle with the mountain as the vertical leg and the distance to the cabin as the horizontal leg. The angle of depression is 46 degrees, which means that the angle between the horizontal leg and the ground is 46 degrees.

  +---------------+
  |               |
  |  46°         |
  |  (angle of   |
  |   depression) |
  |               |
  +---------------+
  |               |
  |  (vertical   |
  |   height of  |
  |   mountain)  |
  |  2400 ft     |
  |               |
  +---------------+
  |               |
  |  (horizontal |
  |   leg)       |
  |  (distance   |
  |   to cabin)  |
  |               |
  +---------------+

Solving for the Indicated Value

Now that we have a picture, we can use trigonometry to solve for the distance to the cabin. We know the vertical height of the mountain (2400 feet) and the angle of depression (46 degrees). We can use the sine function to relate the angle, the vertical height, and the distance to the cabin.

sin(46°) = (vertical height) / (distance to cabin)

We can rearrange this equation to solve for the distance to the cabin:

distance to cabin = (vertical height) / sin(46°)

Now we can plug in the values we know:

distance to cabin = 2400 ft / sin(46°)

Using a calculator to find the sine of 46 degrees, we get:

sin(46°) ≈ 0.7135

Now we can plug this value back into the equation:

distance to cabin ≈ 2400 ft / 0.7135

Solving for the distance to the cabin, we get:

distance to cabin ≈ 3361 ft

Therefore, Dillon's cabin is approximately 3361 feet from the base of the mountain.

Conclusion

In this problem, we used trigonometry to solve for the distance to Dillon's cabin. We drew a picture of the situation, identified the relevant trigonometric function (sine), and used it to relate the angle, the vertical height, and the distance to the cabin. By plugging in the values we knew and solving for the distance, we were able to find the answer.

Key Concepts

Trigonometry
Sine function
Right triangle
Angle of depression
Vertical height
Horizontal leg

Practice Problems

If the vertical height of the mountain is 3000 feet and the angle of depression is 30 degrees, how far is the cabin from the base of the mountain?
If the vertical height of the mountain is 2000 feet and the angle of depression is 60 degrees, how far is the cabin from the base of the mountain?

Solutions

Problem 1:

distance to cabin = (vertical height) / sin(30°)
distance to cabin = 3000 ft / sin(30°)
distance to cabin ≈ 3000 ft / 0.5
distance to cabin ≈ 6000 ft

Problem 2:

distance to cabin = (vertical height) / sin(60°)
distance to cabin = 2000 ft / sin(60°)
distance to cabin ≈ 2000 ft / 0.866
distance to cabin ≈ 2290 ft

References

Trigonometry textbook
Online trigonometry resources
Calculator with trigonometric functions
Directions: Draw a Picture for the Situation, Then Solve for the Indicated Value ===========================================================

Problem 12: Finding the Distance to Dillon's Cabin

Situation

Drawing a Picture

  +---------------+
  |               |
  |  46°         |
  |  (angle of   |
  |   depression) |
  |               |
  +---------------+
  |               |
  |  (vertical   |
  |   height of  |
  |   mountain)  |
  |  2400 ft     |
  |               |
  +---------------+
  |               |
  |  (horizontal |
  |   leg)       |
  |  (distance   |
  |   to cabin)  |
  |               |
  +---------------+

Solving for the Indicated Value

sin(46°) = (vertical height) / (distance to cabin)

We can rearrange this equation to solve for the distance to the cabin:

distance to cabin = (vertical height) / sin(46°)

Now we can plug in the values we know:

distance to cabin = 2400 ft / sin(46°)

Using a calculator to find the sine of 46 degrees, we get:

sin(46°) ≈ 0.7135

Now we can plug this value back into the equation:

distance to cabin ≈ 2400 ft / 0.7135

Solving for the distance to the cabin, we get:

distance to cabin ≈ 3361 ft

Therefore, Dillon's cabin is approximately 3361 feet from the base of the mountain.

Q&A

Q: What is the angle of depression?

A: The angle of depression is the angle between the horizontal leg and the ground. In this problem, the angle of depression is 46 degrees.

Q: What is the vertical height of the mountain?

A: The vertical height of the mountain is 2400 feet.

Q: How do we use trigonometry to solve for the distance to the cabin?

A: We use the sine function to relate the angle, the vertical height, and the distance to the cabin. We can rearrange the equation to solve for the distance to the cabin.

Q: What is the distance to the cabin?

A: The distance to the cabin is approximately 3361 feet.

Q: What is the horizontal leg of the right triangle?

A: The horizontal leg of the right triangle is the distance to the cabin.

Q: What is the sine of 46 degrees?

A: The sine of 46 degrees is approximately 0.7135.

Q: How do we find the sine of an angle?

A: We can use a calculator to find the sine of an angle.

Q: What is the reference angle for 46 degrees?

A: The reference angle for 46 degrees is 90 - 46 = 44 degrees.

Q: What is the sine of 44 degrees?

A: The sine of 44 degrees is approximately 0.6947.

Q: How do we use the reference angle to find the sine of 46 degrees?

A: We can use the reference angle to find the sine of 46 degrees by using the identity sin(90 - x) = cos(x).

Practice Problems

If the vertical height of the mountain is 3000 feet and the angle of depression is 30 degrees, how far is the cabin from the base of the mountain?
If the vertical height of the mountain is 2000 feet and the angle of depression is 60 degrees, how far is the cabin from the base of the mountain?

Solutions

Problem 1:

distance to cabin = (vertical height) / sin(30°)
distance to cabin = 3000 ft / sin(30°)
distance to cabin ≈ 3000 ft / 0.5
distance to cabin ≈ 6000 ft

Problem 2:

distance to cabin = (vertical height) / sin(60°)
distance to cabin = 2000 ft / sin(60°)
distance to cabin ≈ 2000 ft / 0.866
distance to cabin ≈ 2290 ft

References

Trigonometry textbook
Online trigonometry resources
Calculator with trigonometric functions