Optimization Problem Examples
1. Suppose we have 300 feet of fence with which to enclose a rectangular garden on the side
of a house. If the house is used as one side of the garden, how large an area can we
enclose? What is that maximum area?
2. A cylindrical can is to be made to hold 2 Liters of oil. Find the dimensions that will
minimize the cost of the metal to manufacture the can. (That is, give the radius and height
of the cylinder which minimizes the cost of construction materials).
3. Find the point on the parabola 𝑦 " = 2𝑥 that is closest to the point (1, 4).
4. Find the dimensions and area of the largest rectangle that can be inscribed in a semicircle (that is, half-circle) of radius 7.
5. (See below diagram) A man sits in a boat at point A on the riverbank and wishes to get to
point B downstream on the other side of the river. Rowing in a linear path across the
river, he can arrive at point B using one of the following 3 options:
- He can row directly across to C, then run south to B
- He can row in a linear path to somewhere along the opposite bank (point D),
then run the remaining distance down to B
- He can row in a linear path down to B directly
If he can row at rate 6 km/hr and run at rate 8 km/hr, where should he land the boat so he
arrives at B in the least amount of time? (Assume the river water is still).
Optimization Various Practice
1. Find two numbers whose sum is 60 and whose product is a maximum.
2. Find two numbers whose difference is 100 and whose product is a minimum.
3. Find two positive numbers whose product is 60 and whose sum is a minimum.
4. The perimeter of a rectangle is 50 feet. Find the dimensions of the rectangle that encloses the largest
area.
5. Find the dimensions of the rectangle of greatest area that has its base on the x-axis and its other two
vertices above the x-axis and lying on the parabola 𝑦 = 12 − 𝑥 " .
6. Find the point on the line 𝑦 = 2𝑥 + 3 that lies closest to the origin.
7. A manufacturer wishes to design an open box having a square base (no top) and a volume of 32000
cubic feet. What dimensions will produce the box that uses the least amount of material (that is,
minimizes the surface area)?
8. Find the point on the graph of 𝑦 = 𝑥 that is closest to the point 3, 0 .
9. Find the point(s) on the graph of 𝑦 = 4 − 𝑥 " that are closest to the point (0, 2).
10. A container in the shape of a right circular cylinder with no top has a surface area of 3𝜋 square feet.
Find the height and radius of the cylinder with the maximum volume.
11. Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for
the square and how much should be used for the circle to enclose the maximum total area?
12. A box with no top and square base is to be made by cutting equal squares from the corners of a 12 by 12
ft square piece of cardboard and folding up the sides. Find the dimensions of the box with maximum
volume.
13. The container described in exercise #7 is to have volume 10 cubic meters and the length of the base is to
be twice the width (instead of a square base). Material for the base costs $10 per square meter and
material for the sides costs $16 per square meter. Find the cost of materials for the cheapest such
container.
14. A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length
of the shortest ladder that will reach from the ground over the fence to the wall of the building?
15. Find the equation of the line through the point (3 , 5) that cuts off the least amount of area from the first
quadrant.