Find the height of the tree. Show your work algebraically.

Steps to Solve

1. Understanding the Geometry
   We have two triangles involved: the triangle formed by the tree and the ground, and the triangle formed by the wall and the ground.

2. Setting Up the Right Triangle for the Tree
   The height of the tree is represented as $h$, and the distance from the foot of the tree to the point on the ground where the top of the tree’s shadow lies is 24 ft.

3. Setting Up the Right Triangle for the Wall
   In the right triangle formed by the wall, the height of the wall is 4 ft and its distance from the base to the point directly below the top of the wall is 6 ft.

4. Using the Tangent Function
   For the wall triangle, the tangent of the angle $\theta$ can be expressed as: $$ \tan(\theta) = \frac{4 \text{ ft}}{6 \text{ ft}} $$

5. Calculating the Tangent Ratio
   We calculate the tangent: $$ \tan(\theta) = \frac{2}{3} $$

6. Finding the Height of the Tree
   Using the tangent principle for the tree: $$ \tan(\theta) = \frac{h}{24 \text{ ft}} $$ Thus, $$ h = 24 \text{ ft} \cdot \tan(\theta) $$

7. Substituting the Tangent Value
   Now substitute $\tan(\theta) = \frac{2}{3}$: $$ h = 24 \text{ ft} \cdot \frac{2}{3} $$

8. Calculating $h$
   Calculating the height: $$ h = 24 \cdot \frac{2}{3} = 16 \text{ ft} $$