1. **State the problem:** We have two trees 20 m apart horizontally. The first tree is 12 m tall. From the top of the first tree, the angle of elevation to the top of the second tree is 30° and the angle of depression to the base of the second tree is 40°. We need to find the height of the second tree. 2. **Set up the diagram and variables:** Let the height of the second tree be $h$. Let the horizontal distance between the trees be $20$ m. 3. **Analyze the angles and distances:** From the top of the first tree (height 12 m), looking to the base of the second tree with angle depression 40°, the line of sight forms a 40° angle downwards. Using trigonometry, if we drop a vertical from the first tree top to the horizontal line connecting the two trees, the horizontal base distance is adjacent side. 4. **Calculate the horizontal distance from the first tree's top point to the second tree's base:** Using angle of depression 40°: $$ \tan 40^\circ = \frac{\text{vertical drop}}{\text{horizontal distance}} = \frac{12 - 0}{x} = \frac{12}{x} $$ Solve for $x$: $$ x = \frac{12}{\tan 40^\circ} $$ Calculate $\tan 40^\circ$ approximately: $$ \tan 40^\circ \approx 0.8391 $$ So, $$ x = \frac{12}{0.8391} \approx 14.3 \text{ m} $$ 5. **Find the horizontal distance from the first tree's top to the second tree's top:** Since total horizontal distance between trees is 20 m, the distance from the first tree's top to second tree's top is: $$ 20 + \text{any horizontal offset} $$ But this is tricky because the angle of elevation 30° refers to the line from the first tree's top to the second tree's top. The horizontal distance from the first tree's top to the second tree along the ground is: $$ 20 + (x - 20) = x + (20 - x) = 20 $$ Actually, the horizontal distance along the ground is 20 m. But the distance from the first tree top's vertical line to the second tree's vertical line is 20 m. 6. **Let $d$ be the horizontal distance from the first tree's top vertical line to the second tree's base:** From the first tree top, the horizontal distance to the base of the second tree is $x = 14.3$ m (calculated). The remaining horizontal distance to the top of the second tree is: $$ 20 - x = 20 - 14.3 = 5.7 \text{ m} $$ 7. **Calculate the height difference between the tops of the two trees using 30° angle of elevation:** At angle 30°, the tangent is: $$ \tan 30^\circ = \frac{h - 12}{20} $$ Because the horizontal distance between trees is 20 m, and the line from the first tree top to second tree top forms 30° elevation. Calculate $h$: $$ h - 12 = 20 \times \tan 30^\circ $$ $$ \tan 30^\circ \approx 0.5774 $$ So, $$ h - 12 = 20 \times 0.5774 = 11.55 $$ $$ h = 12 + 11.55 = 23.55 \text{ m} $$ 8. **Final answer:** The height of the second tree is approximately **23.55 meters**.