Rescue!! The sky is blue, the sun is hot......you are lying on the beach, almost asleep, when suddenly you become aware of a commotion. Someone is in trouble in the water! Since there is no-one else around and you have recently obtained your life-saving award, it is up to you to go to the rescue. But what is the fastest way of reaching them? Quickly you review the situation: They are 80m along the beach from you and 60m from the shore You can swim at 2 m/s You can run at 5 m/s. How can you reach them in the shortest possible time?

60m 80m ? m

Find the dimensions of the rectangle of perimeter 120cm that has the largest area. Find the coordinates of P that maximise the area of the rectangle shown in the figure below.

An oil field, O, is 10km from shore. The nearest point on land is at S. This would have been a good place to build a refinery except for the fact that there is already a refinery at R, 10km further down the coast. It is planned to build a pipeline running in a straight line under water to some point P on the coast, between S and R and then run it along the shore to R. It costs $1 million per km to build pipeline on land and $2 million per km under water. How far from S should the point P be located in order to minimize the cost of laying the pipe? Pipe Line

SRP O 10km ? km

PROBLEM 1 : Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum. PROBLEM 2 : Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen ? PROBLEM 3 : An open rectangular box with square base is to be made from 48 m 2 of material. What dimensions will result in a box with the largest possible volume ?