1          Find two nonnegative numbers whose sum is 4 and whose product of twice one number and the square of the other is a maximum.

 

2         Build a corral with 3 separate pens (as seen below) using 1000 feet of fencing. What dimensions will maximize the total area of the corral?

 

 

Answer the same questions as above if the corral has 4 pens.  What about if it has n pens?  Describe physically what is happening.

Finally, conjecture the dimensions of the corral with maximum area (still using 1000 feet of fence) if the corral was divided into m rows and n columns?  Below is an

example with 3 rows and 4 columns.   Prove that your conjecture is correct.

 

3         An open rectangular box with square base is to be made from 36 ft.² of material. What dimensions will result in a box with the largest possible volume ?

 

4         A container in the shape of a right circular cylinder has volume 12 ounces What height h and base radius r will minimize the surface of the cylinder ?  (hint: each ounce is about 1.8 cubic inches)

 

5         A sheet of cardboard 4 meters by 5 meters will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume?

 

6         Consider the triangles formed in quadrant I by lines passing through the point (4, 10), the x-axis, and the y-axis. Find the dimensions of the triangle with the shortest hypotenuse.

 

7         Find the point (x, y) on the graph y = x²  closest to the point (3,0).  Find the slope of the tangent line through (x, y).  Find the slope of the line through (x, y) and (3,0).  Conjecture another way to solve this problem based on this observation.  Does your technique work?

 
 

8         Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the window is 12 feet, what dimensions will result in the window having largest possible area?

 

9     The Stupid Phone company makes 40 different kinds of cell phones.  Each month they sell 700 of each type of phone.  For each additional cell phone that Stupid Phone markets, the sales of the other phones decrease by 10. How many new phones should Stupid Phone market to maximize its total monthly sales of phones?

 

 

10     Find the dimensions of the rectangle of largest area which can be inscribed inside a circle of radius 9.
 

11     Draw an isosceles triangle with two sides of length 6. What is the length of the base that will give your triangle the largest area?

 

12     Find the point(s) (x,y) on the graph of  y = (1+x² )^-1 where the tangent lines to the graph at (x,y) maximum slope.

 

13  Find the length of the shortest ladder that will pass over a 10-ft. high barrier to a reach a very high wall which is 4 ft. behind the fence. (See diagram.)

14     Car A is traveling due north toward Sausalito at 60 mph.  Car B traveling due west toward Sausalito at 70 mph. At 2:00 Cars A and B are 100 miles and 120 miles from Sausalito respectively.  At what time is the distance between the cars a minimum?  What is the minimum distance?