1. **State the problem:** A forest ranger at the top of a 45-foot fire tower sees his partner on the ground at an angle of depression of 40°. We need to find the horizontal distance from the base of the tower to the partner. 2. **Understand the angle of depression:** The angle of depression from the top of the tower to the partner is 40°. This angle is equal to the angle of elevation from the partner to the top of the tower due to alternate interior angles formed by the horizontal line. 3. **Identify the right triangle:** - The vertical side (height of the tower) is 45 feet. - The angle adjacent to the horizontal side is 40°. - The horizontal side (distance from the base to the partner) is unknown, call it $x$. 4. **Use the tangent function:** Tangent relates the opposite side to the adjacent side in a right triangle: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ Here, $\theta = 40^\circ$, opposite side = 45 feet, adjacent side = $x$. 5. **Set up the equation:** $$\tan(40^\circ) = \frac{45}{x}$$ 6. **Solve for $x$:** Multiply both sides by $x$: $$x \times \tan(40^\circ) = 45$$ Divide both sides by $\tan(40^\circ)$: $$x = \frac{45}{\tan(40^\circ)}$$ Show cancellation: $$x = \frac{45}{\cancel{\tan(40^\circ)}} \times \frac{1}{\cancel{\tan(40^\circ)}}$$ 7. **Calculate the value:** Using a calculator, $$\tan(40^\circ) \approx 0.8391$$ So, $$x = \frac{45}{0.8391} \approx 53.63$$ 8. **Final answer:** The partner is approximately **53.63 feet** from the base of the tower.