1. **State the problem:** We want to analyze the function $$T = \frac{\sqrt{2000 - 80x + x^2}}{2}$$ and related expressions, which represent distances involving a lifeguard, a point on the beach at distance $x$, and a swimmer in the water. 2. **Rewrite and interpret the expressions:** - The term $$\sqrt{2000 - 80x + x^2}$$ can be recognized as a distance formula. Notice that $$2000 - 80x + x^2 = (x - 40)^2 + 20^2$$ because: $$ (x - 40)^2 + 20^2 = (x^2 - 80x + 1600) + 400 = x^2 - 80x + 2000 $$ 3. **Simplify the first expression:** $$ T = \frac{\sqrt{(x - 40)^2 + 20^2}}{2} $$ This represents half the distance from the point $x$ on the beach to the swimmer located 20 meters deep in the water and 40 meters along the beach. 4. **Analyze the second expression:** $$ T = \frac{\sqrt{20^2 + x^2}}{2} $$ This is half the distance from the lifeguard (20 meters above the beach) to the point $x$ meters along the beach. 5. **Analyze the combined expression:** $$ T = \frac{1}{6} \sqrt{20^2 + x^2} + \frac{1}{2} \sqrt{2000 - 80x + x^2} $$ This represents a weighted sum of the distances from the lifeguard to point $x$ on the beach and from point $x$ to the swimmer. 6. **Summary of the geometry:** - The lifeguard is 20 meters above the beach. - The swimmer is 20 meters deep in the water and 40 meters along the beach from the lifeguard's vertical position. - The variable $x$ represents a point along the beach between 0 and 40 meters. - The distances are calculated using the Pythagorean theorem. 7. **Final note:** These expressions model the distances involved in the lifeguard's path to the swimmer via a point $x$ on the beach, useful for optimization problems such as minimizing rescue time.