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We are given the following information:

- The student is standing 20 feet away from the base of the tree.
- The angle of elevation to the top of the tree is 50°.
- The student's eyes are 5 feet above the ground.
- We are asked to find the height of the tree, using \( \cos 50^\circ \approx 0.64 \).

Let's break it down:

1. The triangle formed between the student, the base of the tree, and the top of the tree is a right triangle. 
2. The horizontal distance between the student and the base of the tree is 20 feet, which forms the adjacent side of the triangle.
3. We need to calculate the total height of the tree (which includes the height from the student's eyes and the additional height from the top of the tree to his line of sight).

Using the cosine formula for the right triangle:
\[
\cos(50^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Where:
- \(\cos(50^\circ) \approx 0.64\)
- The adjacent side is 20 feet.

Let’s calculate the hypotenuse (the slant height from the student's eyes to the top of the tree):
\[
\cos(50^\circ) = \frac{20}{\text{hypotenuse}}
\]
\[
\text{hypotenuse} = \frac{20}{0.64}
\]
Now let's compute the hypotenuse value.The hypotenuse (the slant height from the student's eyes to the top of the tree) is approximately 31.25 feet.

Now, using the sine function, we can calculate the opposite side (the vertical height from the student's eye level to the top of the tree):

\[
\sin(50^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\text{opposite} = \sin(50^\circ) \times \text{hypotenuse}
\]
Next, we’ll calculate the sine of 50° and solve for the opposite side. Finally, we'll add the height from the student's eyes (5 feet) to find the total height of the tree.Since we are not given \( \sin(50^\circ) \) directly, we can first estimate it using trigonometric identities or find it approximately. 

For now, though, we can use the relationship between the sides:

- The cosine formula only gives us horizontal distances relative to the angle, so instead, we can recognize the adjacent side and calculate using tangent or sine. Would you like to confirm this

A student is standing 20 feet away from the base of a tree. He looks to the top of the tree at a 50° angle of elevation. His eyes are 5 feet above the ground. Using cos 50° ≈ 0.64, what is the height of the tree to the nearest foot?

Learn how to calculate the height of a tree using trigonometry. Given a 50° angle of elevation and a distance of 20 feet from the base, this problem explores the use of the cosine function to determine the height. Ideal for high school students in grades 9-11.

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