MATH 100 V1A
November 14th – Practice problems
Hints and Solutions
1. Speculate the volume of the largest rectangular box which may be inscribed in a sphere.
Solution: Since we already saw that the largest rectangle that can be inscribed inside
a circle is a square, it seems plausible for the largest rectangular box which can be
inscribed in a sphere to be a cube (this turns out to be the case – you’ll learn how
to prove this in a multivariable calculus course). If the sphere has radius r, then the
length of each side of the cube will be √13 r, so the volume is 3√1 3 r3 .
2. Suppose a spherical cell has a rate of nutrient absorption which is proportional to its
surface area, and a rate of nutrient consumption which is proportional to its volume.
Let the constants of proportionality be a and b respectively. What is the optimal radius
of the cell?
Hint: Come up with an equation which describes the overall rate of change nutrient
levels in the cell (the rate of absorption minus the rate of consumption) with respect
to time. Think of this as a function of r (note that r does not depend on t) and find
the r which maximizes this rate.
3. Suppose you wish to connect the four corners of the unit square with a path (which
may have branches). Make a√conjecture as to which path is the shortest. (Hint: The
shortest path has length 1 + 3).
Hint: Notice that the red path is slightly shorter than the black path (below) because
we only have to count the horizontal piece once.
1 y
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x
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1