Let's break down the problem and solve it step by step.

Step 1: Understanding the Setup

• You're standing on one side of the river.
• There is a tree on the opposite bank, and the tree is 150 feet tall.
• A line from the top of the tree to the ground at your feet makes an angle of 11° with the ground.
• We need to find how far you are from the base of the tree, i.e., the distance across the river.

Step 2: Sketch the Problem

Here's a description of the situation:

1. The tree forms a vertical line with a height of 150 ft.
2. The angle of elevation from your point (on the bank of the river) to the top of the tree is 11°.
3. The distance between you and the base of the tree is the horizontal distance we need to find.

This setup forms a right triangle where:

• The tree's height (150 ft) is the opposite side.
• The distance from you to the base of the tree is the adjacent side.
• The angle of elevation is 11°.

Step 3: Applying Trigonometry

We can use the tangent function, which relates the opposite side, adjacent side, and angle in a right triangle:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

In our case:

• $\theta = 11^\circ$
• The opposite side (tree height) = 150 ft
• The adjacent side (distance from you to the base of the tree) = $x$

So, we can set up the equation:

$\tan(11^\circ) = \frac{150}{x}$

Now, solve for $x$:

$x = \frac{150}{\tan(11^\circ)}$

Step 4: Calculating the Distance

Let's calculate the value of $x$.

Using a calculator for $\tan(11^\circ)$:

$\tan(11^\circ) \approx 0.1944$

So:

$x = \frac{150}{0.1944} \approx 771.7$

Step 5: Rounding the Answer

The distance is approximately 772 feet.

Final Answer:

The base of the tree is approximately 772 feet away from you.

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Would you like further details or have any questions on the calculation?

Here are 5 related questions to explore:

1. How can we use the sine or cosine functions in this scenario instead of tangent?
2. What happens if the angle of elevation increases to 20°? How would that affect the distance?
3. How can we calculate the height of the tree if the angle of elevation and the distance were given instead?
4. What would be the angle of elevation if you were standing 500 feet away from the base of the tree?
5. How does this type of trigonometric problem relate to real-world navigation and surveying?

Tip: When dealing with angles in trigonometry, always make sure your calculator is set to the correct mode (degrees or radians) based on the problem.