Problem 8: 1998-1999
A company sells life size cardboard cutouts of Richard Moll (the
adorable, fun-loving, and clearly bald "Bull" from TV's Night Court) for which
the number of units sold, S in millions, is
,
where t is time in months.
(a) Find the average rate of change of S(t) during the
first year.
(b) During what month does S'(t) equal the average value
during the first year? (The value guaranteed by the Mean Value Theorem.)
(c) Would the French club make more money selling life
size "Bulls" than they could with those chocolate advent calenders?
Solution:
(a) We need to figure out S(0) and S(12) to determine
what the average rate of change is. Remember that the formula is
.
If you plug 0 into S, you get 100. I guess this means they
had some sales before the year started; that's okay. Now, it's easy to figure
out that S(12) = 871.4285714, which is 6100/7. Plug into the above formula
to get 64.286 Bulls/month as the average rate of change.
(b) Now, we want to figure out when exactly S'(t) =
64.286. The derivative is going to be
.
Set this equal to 64.286 and cross multiply to get a really really
ugly quadratic equation. After much pain and suffering, we get a solution of
approximately 2.89. If t = 0 represents the beginning of the year and
t = 12 represents the end of December, 2.89 means sometime toward the
end of February.
(c) Hard to say, but a chocolate Bull
would sell the best.
