Prepared : Wed Oct 20 21:56:35 CDT 1999
MATH 1271
QUIZ IV
1) Three large squares of tin, each with edges 1 m long, have four small, equal squares cut from their corners. All twelve resulting small squares are to be the same size (Figure below). The three large cross-shaped pieces are then folded and welded to make boxes with no tops, and the twelve small squares are used to make two small cubes. How should this be done to maximize the total volume of all five boxes?
Each of the boxes with no tops will have
The small cubes will be made from the small cut parts, hence their volume will be (each)
There are three open boxes and two cubes, hence the total volume is
To find the critical points, first differentiate
with respect to x.
The derivative is zero when
There are also endpoints which have to be considered. In our case the
endpoints will be 0 and 1/2, since x denotes a length (which is not
negative) and 2x 
1.
Plug all the critical points and the endpoints into
and see that when
x=1/2 we have the greatest .
2) A hiker starting at a point P on a straight road
wants to reach a forest cabin that is 2 km from a point Q 3 km down the
road from P (Figure below). She can walk 8 km/h along the road but only
3 km/h through the forest. She wants to minimize the time required to
reach the cabin. How far down the road should she walk before setting
off through the forest straight for the cabin? (Suggestion: Use the
angle 
as the independent variable.)
The total time spent will be the sum of the time spent on the road and the time spent in the forest. We can express these quantities in pseudo-math as follows:
The hint in the problem tells us to use as the variable. Call
the amount walked in the forest by y, and the amount walked on the road
by x. Then
which gives that 
.
Similarly we get that
so 
.
Now we can change from the pseudo-math to real math in our first formula.
Differentiate Time with respect to .
The critical point will be when
that is when
But actually the problem is not asking for , rather we are asked
to find x.
