Question

Guillermo is a professional deep water free diver.
His altitude (in meters relative to sea level),
\[x\] seconds after diving, is modeled by

\[g(x)=\dfrac{1}{20}x(x-100)\]
How many seconds after diving will Guillermo reach his lowest altitude?
seconds

Answer 1

To find out when Guillermo reaches his lowest altitude, we need to identify the vertex of the quadratic function given by

\[
g(x) = \dfrac{1}{20} x(x - 100).
\]

This function is a quadratic equation in the standard form \(g(x) = ax^2 + bx + c\). We can rewrite the function as:

\[
g(x) = \dfrac{1}{20} (x^2 - 100x) = \dfrac{1}{20} x^2 - \dfrac{5}{1} x.
\]

Here, \(a = \dfrac{1}{20}\) and \(b = -5\).

The time \(x\) at which the vertex of a parabolic function occurs (which gives either the maximum or minimum, depending on the leading coefficient) can be found using the formula:

\[
x = -\frac{b}{2a}.
\]

Substituting the values of \(a\) and \(b\):

\[
x = -\frac{-5}{2 \cdot \frac{1}{20}}.
\]

Calculating this gives:

\[
x = \frac{5}{\frac{1}{10}} = 5 \times 10 = 50.
\]

Thus, Guillermo reaches his lowest altitude **50 seconds** after diving.