MCV 4U Calculus and Vectors
Optimization Problems
Use your knowledge of the calculus to solve the following problems.
Write full solutions. Remember in each case to justify the classification
of your optimal point.
Problem 1
Patty is going to build a window of perimeter 33 units that consists of a
rectangle surmounted by an equilateral triangle. What should the width
of the window be to admit the most sunlight?
Problem 2
A rectangular poster contains 5400 cm2. The margins at the top and
bottom are each 3 cm and the margins at the sides are each 2 cm. What
are the dimensions of the poster if the printing area is a maximum?
Problem 3
A page measures 18 cm by 24 cm. The lower left corner is folded over
to the opposite right-hand edge of the page. If the length of the crease is
as short as possible, find the width of the part folded over, measured
along the bottom of the page.
Problem 4
During coughing, the diameter of the trachea decreases to increase the
velocity of the air going out. The velocity v of air in the trachea during a
cough is related to the radius of the trachea during the cough by the
𝑟
equation 𝑣 = 𝐴𝑟 2 (𝑟0 − 𝑟) where 0 ≤ 𝑟 ≤ 𝑟0 . In this equation, A is a
2
positive constant and 𝑟0 is the radius of the trachea in a relaxed state.
Find the radius of the trachea when the velocity is greatest, and find
maximum velocity of air.
Problem 5
Two hallways 3 m wide and 2 m wide meet at right angles. A heavy pole
is to be pushed on dollies (i.e. horizontally) around the corner.
Disregarding the pole’s diameter, what is the maximum length of pole
that will be able to make the turn?
Problem 6
A fence 2 m high is 3 m from a wall. Find the length of the shortest
plank ht, while resting on the ground, can pass over the fence to brace
the wall.
Problem 7
Marcy wants to cut a 10-cm wire into two parts and form one piece into
a circle and the other into a square. How should she cut the wire so that
the total area of the two shapes is a minimum?
Problem 8
Bob sits in a rowboat 4 km from a straight shoreline. He wants to go to a
store that lies 6 km down the shore from the point on the shoreline
closest to Bob. He plans to row to a point P on the shoreline and walk
the remaining distance to the store. Bob rows 3 km/h and walks 5 km/h.
To what point should he row if he wants to minimize his travelling time?
Answers
1. 6 + √3 units
2. 60 cm by 90 cm
3. 13.5 cm along
4. 𝑟 =
2
2
3
𝑟0 and 𝑣 =
4
27
𝐴𝑟03
3
2 2
3
5. (23 + 3 )
6. ???
7.
10𝜋
4+ 𝜋
cm for the circle
8. 3 km along the shore