Steps for Solving Optimization Problems

  1. Draw the picture and label variables.

  2. Determine a constraint equation  (if necessary) and a maximizing (minimizing) equation.

  3. Use the constraint equation to solve for one of the variables and substitute it into the maximizing (minimizing) equation.

  4. Take a derivative and set it equal to zero.  Then solve.

  5. Answer the question.



  1. You want to construct a can that holds 150 cubic inches of juice as cheaply as possible. The top and bottom costs .1 cents per square inch and the side costs .09 cents per square inch. What should the dimensions of the can be?


    First we use the volume of a cylinder to get the constraint equation

    150 = pr2h 

    The cost equation gives us our optimization equation 

            Cost = 2pr2(.1) + 2prh(.09)        Cost of top and bottom + Cost of sides

    The volume equation gives us:

            h = 150/pr2  

    so that

            C = .2pr2 + .18pr(150/pr2)

            = .2pr2 + 27/r

    To find the minimum cost we take the derivative and set it equal to 0:

            C' = 4pr - 27/r2 = 0

    So that

            4pr3 = 27


            r3 = 27/(4p)

            r = 2.14 in

    so that 

            h = 150/p(2.142) =10.4 in

  2. A lifeguard swims at a rate of 5 feet per second and can run at a rate of 15 feet per second. Suppose that the lifeguard spots a drowning child in the ocean 200 feet down the shore and 50 feet out at sea. How far should the lifeguard run until (s)he begins swimming?


    Our goal is to minimize the total transit time.  The total transit time is 

            Total Transit Time (T) = Time Along the Beach + Time in the Water


            Time = Distance/Rate

    We have 

            TimeBeach = x/15


            TimeWater =


    Taking a derivative and setting it equal to 0 gives


    After a lot of algebra or using a computer we get

            x = 200 - 25/ @ 182.3 feet

    We can conclude that the runner should run a little more than 182 feet before diving into the water.



Hold your mouse over the yellow rectangle for the answer.

  1. A poster is to have an area of 120 square inches with one inch margins at the bottom and sides and a 2 inch margin at the top.  What dimensions will give the largest printed area?        6.32 by 18.97

  2. A quarter mile race track is to be designed by having a rectangle with semicircles on each end.  Find the dimensions that will make the area of the rectangle as large as possible.        1/(8pi) by 1/8


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