On the way to the mall Miguel rides his skateboard to get to the bus stop. He then waits a few minutes for the bus to come, then
rides the bus to the mall. He gets off the bus when it stops at the mall and walks across the parking lot to the closest entrance. Which graph correctly models his travel time and distance?
2 answers:
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Notice that
so the constraint is a set of two lines,
and only the first line passes through the first quadrant.
The distance between any point in the plane is
, but we know that
and
share the same critical points, so we need only worry about minimizing
. The Lagrangian for this problem is then
with partial derivatives (set equal to 0)
We have
which tells us that
so that is a critical point. The Hessian for the target function
is
which is positive definite for all , so the critical point is the site of a minimum. The minimum distance itself (which we don't seem to care about for this problem, but we might as well state it) is
.