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CALCULUS BC PROJECT
CHOCOLATE MAKER PROBLEM
VERSION A
NAME:
DATE:
HOUR:
PROBLEM PRESENTED:
A chocolate maker wants to package chocolate pieces in boxes each having a volume of 1000
cubic units. Each of these chocolate boxes is to be an open-topped rectangular box with a
square base having edge length x between 6 and 12 units. In addition, the box is also to have a
square lid with a 2-unit rim. Thus the box-with-lid actually consists of two open-topped boxes:
the x by x by y candy box itself with height y  2 and the x by x by 2 lid where the height is 2
units. (We will assume the lid fits VERY tightly for simplicity reasons.) The chocolate maker’s
problem is to determine the dimensions x and y that will minimize the total area A (and hence
the cost) of the two open-topped boxes that comprise a single chocolate box with its lid.
I. SOLVE THE CHOCOLATE MAKER’S PROBLEM:
Solve the chocolate maker’s problem using a preset volume of 1000 cubic units. Begin by
expressing the total area of the two open-topped boxes as a function of A in terms of the base
edge length x. Show that the equation A'(x )  0 simplifies to the cubic equation
x 3  2x 2  1000  0 . (HINT: Factoring out common factors and negative exponents is always
helpful!!) But instead of attempting to solve this equation directly, graph A(x ) and A'(x ) on
the same set of coordinate axes and find the zero(s) of A'(x ) .
II. SOLVE YOUR OWN PERSONALIZED CHOCOLATE BOX PROBLEM:
For your own personal chocolate box problem, suppose that the box itself is to have a volume
where V  400  50n cubic units, where n is a digit assigned by your math teacher. The lid still
has a 2-unit rim. If you make your box-with-lid out of whatever material your prefer where
cost is C  2n  38.5 per square unit, what dimensions will minimize the total cost of the
material needed? What is this minimum cost?
III. SOLVE YOUR OWN PERSONALIZED CHOCOLATE CYLINDER PROBLEM:
Suppose instead that you want to make a cylindrical box as indicated in the figure. Now the
base container and its lid are both open-topped circular cylinders. Find the dimensions r and h
that minimize the cost of this cylindrical container-with-lid (with the same volume
V  400  50n cubic units as in part II and with its lid having a 2-unit rim as before and the
same cost per square unit equation C  2n  38.5 ).
Which costs less to make: the rectangular box or the cylindrical container?