Optimization Problems

Problem 1: Maximum Volume of a Cardboard Box

1. We have a piece of cardboard that is 50 cm by 20 cm and we are going to cut out the corners and fold up the
sides to form a box. Determine the height of the box that will give a maximum volume.
Complete solutions are required
Include a diagram where possible

2. We want to construct a cylindrical can with a bottom but no top that will have a volume of 30cm3 Determine
the dimensions of the can that will minimize the amount of material needed to construct the can.

3. We want to construct a window whose middle is a rectangle and the top of the window is semi-circle. If we
have 50 meters of framing material what are the dimensions of the window that will let in the most light?
(Framing material will be needed to separate the semicircle from the rectangular part of the window).

4. Determine the area of the largest rectangle that can be inscribed in a circle of radius 1.

5. A line through the point (2,5) forms a right triangle with the x-axis and y-axis in the 1st quadrant. Determine
the equation of the line that will minimize the area of this triangle.

6. A ladder is being carried down a hallway that is 15 feet wide. At the end of the hallway there is a right-angled
turn and the hallway narrows down to 9 feet wide. What is the longest ladder (always keeping it horizontal)
that can be carried around the turn in the hallway?

7. Two 10-meter-tall poles are 30 meters apart. A length of wire is attached to the top of each pole and it is
staked to the ground somewhere between the two poles. Where the wire should be staked so that the
minimum amount of wire is used?

8. We have 45 m2 of material to build a box with a square base and no top. Determine the dimensions of the box that will maximize the enclosed volume.

9. We want to build a box whose base-length is 6 times the base-width and the box will enclose 20m3
The cost of the material of the sides is $3/ m2 and the cost of the top and bottom is $15/ m2 Determine the dimensions of  the box that will minimize the cost. [THINKING/INQUIRY]

10.We are going to fence in a rectangular field. If we look at the field from above the cost of the vertical sides are $10/m, the cost of the bottom is $2/m and the cost of the top is $7/m. If we have $700 determine the dimensions of the field that will maximize the enclosed area.