Honors 2201-4
Homework 13: due Wednesday November 19
(from Applied Calculus, Hughes-Hallett et.al.)
Fall 2003
For all problems and parts, justify and explain your answers and conclusions.
(1) On the west coast of Canada, crows eat whelks (a shell-fish). To open the whelks, the
crows drop them from the air onto a rock. If the shell does not smash the first time, the
whelk is dropped again. The average number of drops, n, needed when the whelk is dropped
from a height of x meters is approximated by
n(x) = 1 +
27
x2
(a) For this example, what is the domain of n(x)?
(b) What happens to the average number of drops required when the height of the drop
becomes very small? What happens to the average number of drops required when the
height of the drop becomes very large?
(c) Express the average total vertical distance the crow travels upward to successfully open
a whelk as a function of drop height, x.
(d) Crows are observed to drop whelks from the height that minimizes the total vertical
upward distance traveled per whelk. What is this height, and what is the average number
of drops for this height?
(2) The quantity of a drug in the bloodstream t hours after a tablet is swallowed is given,
in mg, by
q(t) = 20 e−t − e−2t .
(a) For this example, what is the domain of q(t)?
(b) How much of the drug is in the bloodstream at time t = 0?
(c) When is the maximum quantity of drug in the bloodstream? What is the maximum
quantity?
(d) In the long run, what happens to the quantity of drug in the bloodstream?
(Extra) In general, for what values of t is q(t) > 0? Justify your answer algebraically.
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(3) What is the area of the largest triangle that can be formed in the first quadrant by the
x-axis, the y-axis, and a line tangent to the graph of y = e−x ? The key here is to find the
line tangent to the graph of y = e−x that creates the largest triangle. Remember to specify
the domain and consider endpoints in your calculations.
(4) Your local pizza delivery uses square boxes that are made from rectangular pieces of
corrugated cardboard, each 47 by 90cm. The box is made by cutting out six small squares,
three from each of the 90cm edges, one square at each corner and one in the middle of the
edge. The result is then folded into a box in the obvious fashion. How does one design a box
in this fashion to obtain the largest volume? You might want to start by drawing a picture.
Remember to specify the domain and consider endpoints in your calculations.
(5) In a chemical reaction, substance A combines with substance B to form substance Y .
At the start of the reaction, the quantity of A present is a grams, and the quantity of B
present is b grams. Assume a < b. At time t seconds after the start of the reaction, the
quantity of Y present is y grams. For certain types of reactions, the rate of the reaction, in
grams per second, is given by
R(y) = k(a − y)(b − y),
where R(y) is the rate of the reaction, and k is a positive constant.
(a) For what values of y is the rate nonnegative?
(b) What is the domain for R(y) that makes sense for the chemical reaction?
(c) Find the value of y at which the rate of the reaction is fastest. Remember to use sensible
endpoints.
(6) An apple tree produces, on average, 400kg of fruit each season. However, if more than
200 trees are planted per km2 , crowding reduces the yield by 1kg for each tree over 200.
(a) Express the total yield from one square kilometer as a function of the number of trees
on it. Your function will have to be defined in two parts. Specify the domain for each
part.
(b) How many trees should a farmer plant on each square kilometer to maximize yield?
Remember to consider endpoints.
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