Question: Guillermo is a professional deep water free diver.
His altitude (in meters relative to sea level), $x$ seconds after diving, is modeled by
g(x) = \frac{1}{20} x(x - 100)
What is the lowest altitude Guillermo will reach?
meters relative to sea level
1. **State the problem:** We want to find the lowest altitude Guillermo reaches, given the altitude function $$g(x) = \frac{1}{20} x(x - 100)$$ where $x$ is the time in seconds after diving.
2. **Rewrite the function:** Expand the expression:
$$g(x) = \frac{1}{20} (x^2 - 100x) = \frac{1}{20} x^2 - 5x$$
3. **Identify the type of function:** This is a quadratic function of the form $$g(x) = ax^2 + bx + c$$ with $$a = \frac{1}{20} > 0$$ and $$b = -5$$.
Since $$a > 0$$, the parabola opens upwards, meaning the vertex is a minimum point.
4. **Find the vertex:** The $x$-coordinate of the vertex is given by $$x = -\frac{b}{2a}$$.
Calculate:
$$x = -\frac{-5}{2 \times \frac{1}{20}} = \frac{5}{\frac{2}{20}} = 5 \times \frac{20}{2} = 5 \times 10 = 50$$
5. **Find the altitude at the vertex:** Substitute $$x=50$$ into $$g(x)$$:
$$g(50) = \frac{1}{20} \times 50 \times (50 - 100) = \frac{1}{20} \times 50 \times (-50) = \frac{1}{20} \times (-2500) = -125$$
6. **Interpretation:** The lowest altitude Guillermo reaches is $$-125$$ meters relative to sea level.
**Final answer:** $$\boxed{-125}$$ meters relative to sea level.