 # Advanced Calculus Assignment 6 - Applied Optimization

Assignment 6

Applied Optimization

You must submit solutions for problems 2, 4, and 6 (5 marks each question). Problems 1, 3, and 5 are
provided as practice problems (answers given on the back of this sheet). To get any marks for any of the
bonus questions, your solution must be substantially complete and correct.
1. A rectangular workyard of area 3,000 square feet is to be enclosed. Because of different exposures, the
fence along two opposite sides of this rectangular region will cost \$20.88 per foot, whereas the fence along
the other two opposite sides will cost \$40.88 per foot. Determine the dimensions of this yard which will
minimize the cost of the fencing and compute the total cost of fencing (before tax) for this optimal design.
State your answer as three numerical values in order:
(i) length of yard along the sides that require \$20.88 per foot fencing ,
(ii) length of yard along the sides that require \$40.88 per foot fencing, and finally
(i ii) total dollar cost of fencing for the yard.
Round all of your answers to two decimal places.
2. A large rectangular warehouse (the shaded region in the diagram) is to
be built to satisfy the following conditions. The warehouse must have a
total area of 10600 square meters. At the front and the back of the
warehouse , there must be at least 24 meters of land to the property line to
allow for loading vehicles (the borders labelled 'a' in the diagram to the
rig ht). On the sides of the building, there must be at least 14 meters of land
to the property line (the borders labelled 'b' in the diagram). Determine the
width (distance across the front) and length (distance front to back) of the
warehouse so that the total land area required is as small as possible.
Also calculate that minimum land area required . Yot,Jr answer must be
three numbers in order:
(i) the width of the warehouse in meters,
(ii) the length of the warehouse in meters, and
(iii) the total land area required in square meters.
a
a
3. A closed top cylindrical container is to be made to hold 8.9 litres. The cost of materials for the circular top
and bottom are twice as much as the cost of materials for the cylindrical tube forming the sides. Determine
the dimensions of this container (rand h) to result in the smallest possible cost of materials.
4. A closed top cylindrical container is to be fashioned to have a surface area no larger than 14500 square
centimetres. Determine the dimensions of the container satisfying this condition which will have the largest
possible internal volume.
5. A company must fabricate cylindrical steel containers with a volume of 215 litres
for shipping their product. Because of the nature of their product, the welds required
to form the closed container (the heavy lines in the diagram) are particularly costly,
and so the goal is to determine dimensions for this container which will result in the
smallest possible total cost of producing these welds. The weld around the bottom
of the container is three times as costly as the other welds per centimetre.
Determine what the radius and height of the container must be.
Your answer will consist of two numbers, in order:
(i) the radius of the circular cross-section of the container, and
(ii) the height of the container,
both in centimetres.
6. A piece of wire 320 em long can be formed into a circle, or a square, or it can be cut into two pieces to
form both a circle and a square. Determine the dimensions of the shape or shapes formed which will result
in (a) a maximum combined area being enclosed, and (b) a minimum combined area being enclosed by the
two shapes.
continued over .. ..

Bonus Question:
81 . (a) The diagram to the right is a view from above of
a large service conduit channel that must be constructed
from point B to point A along the ceiling of a large
production room. The section from point A to point C
needs to be suspended from the ceiling, and would cost
\$840 per meter, whereas the stretch from point C to
point B would cost just \$620 per meter because the
conduit can be supported by the wall as well as the
ceiling. Determine the distance from point D to point C
so that the total cost of the conduit is minimized. Also
calculate what that minimum cost would be. The
distance from point B to point D is 52 meters and the
distance from point D to point A is 36 meters. Obviously
the diagram is not necessarily drawn accurately to scale.
Your answer should be two numbers in order:
(i) the distance between points D and C, and
(ii) the actual cost of constructing the entire conduit
based on the above information. (3 marks)

(b) Repeat part (a) but now with the cost of conduit along the wall from C to B being \$710 per meter.
8
(2 marks)
82. Rework problem 1, but represent the costs of the fencing by symbolic constants (say, one pair of
opposite sides cost g dollars per foot, and the other pair of opposite sides cost h dollars per foot).
Represent the area by the constant A. Obtain formulas for the length and width of the work yard which will
minimize the total cost of fence. Then, work out formulas for the total cost of each of the two types of fence,
and make a surprising (or at least perhaps unexpected) observation about the relationship between these
two costs.
Answers to Practice Problems:
(1) (i) 76.64 ft; (ii) 39.14 ft; (iii) \$6400.90
(3) r = 8.914 em; h = 35.655 em
(5) r = 17.594 em; h = 221 .09 em
(4 marks)

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Subjects: Calculus
Level: College / University
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## Calculus Homework Solutions

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