Question

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

\[D(x)=\dfrac{1}{36}(x-60)^2-100\]
What is the lowest altitude Ying will reach?
meters

Answer 1

To find the lowest altitude Ying will reach, we need to analyze the given function:

\[
D(x) = \dfrac{1}{36}(x - 60)^2 - 100
\]

This is a quadratic function of the form \(D(x) = a(x - h)^2 + k\), where:
- \(a = \dfrac{1}{36}\)
- \(h = 60\)
- \(k = -100\)

The vertex of a quadratic function \(y = a(x - h)^2 + k\) gives the maximum or minimum value, depending on the sign of \(a\):
- If \(a > 0\), the vertex represents the minimum point.
- If \(a < 0\), the vertex represents the maximum point.

In this case, since \(a = \dfrac{1}{36} > 0\), the vertex represents the minimum altitude.

The coordinates of the vertex are given by:
- \(h = 60\) (the x-coordinate, which corresponds to the time after diving)
- \(k = -100\) (the lowest altitude)

Thus, the lowest altitude Ying will reach is:

\[
\boxed{-100} \text{ meters}
\]