You hike 190 meters up a steep hill that has a 33° angle of elevation as shown in the diagram

You hike 190 meters up a steep hill that has a 33° angle of elevation as shown in the diagram.

Give each side and angle measure rounded to the nearest whole number.

The Correct Answer and Explanation is :

It seems that I can’t access the image directly. However, I can help you solve the problem using the information you provided about the situation.

Given:

• A hill with a 33° angle of elevation.
• The distance you hike (the hypotenuse) is 190 meters.

You can apply trigonometry to find the lengths of the other sides of the right triangle formed by the hill.

Steps:

1. Identify the triangle’s components:

• The hypotenuse (190 meters) is the distance along the slope of the hill.
• The angle of elevation is 33°.
• You need to find the length of the vertical side (height of the hill) and the horizontal side (base of the hill).

1. Use trigonometric functions:

• Sine relates the opposite side (height of the hill) to the hypotenuse:
  [
  \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
  ]
  So, the height (opposite side) will be:
  [
  \text{height} = 190 \times \sin(33^\circ)
  ]
• Cosine relates the adjacent side (base of the hill) to the hypotenuse:
  [
  \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
  ]
  So, the base (adjacent side) will be:
  [
  \text{base} = 190 \times \cos(33^\circ)
  ]

1. Perform the calculations:

• Using the values for (\sin(33^\circ) \approx 0.544) and (\cos(33^\circ) \approx 0.838):
  • Height:
    [
    \text{height} = 190 \times 0.544 \approx 103.36 \text{ meters} \quad (\text{round to } 103 \text{ meters})
    ]
  • Base:
    [
    \text{base} = 190 \times 0.838 \approx 159.22 \text{ meters} \quad (\text{round to } 159 \text{ meters})
    ]

1. Final answer:

• Height = 103 meters
• Base = 159 meters
• Hypotenuse = 190 meters (as given)
• Angle = 33° (as given)

Explanation:

This is a typical trigonometry problem where you apply the sine and cosine functions to find the height and base of a right triangle when the hypotenuse and one angle are known. By using these functions, we can break the sloped distance into its vertical and horizontal components, helping us understand the height climbed and the distance along the ground.

Let me now generate the corresponding diagram for you!

Here is the diagram of the right triangle, showing the steep hill with the given measurements. This visualization should help clarify the relationship between the sides and the angle of elevation. Let me know if you need further assistance!