Answer the following question

Modelling and investigation activity The problem You are standing at the edge of a slow-moving river which is one kilometre wide. You want to return to your campground on the opposite side of the river. You can swim at 3 km/h and run at 8 km/h. You must first swim across the river to any point on the opposite bank. From there you must run to the campground, which is 2 km from the point directly across the river from where you start your swim. What route will take the least amount of time? Visualize the problem Here is a diagram of this situation: 2KM (B ) X (C ) 2 - X CAMP GROUND (D) 1 KM RIVER You (A) Discuss what each label in the diagram represents. 274Solve the problem What is the length of AC in terms of x? Using the formula for time taken for travel at a constant rate of speed from this chapter, write down an expression in terms of x for: 1 the time taken to swim from A to C 2 the time taken to run from C to D. D in terms of x. Hence write down an expression for the total time taken, T, to travel from A to You want to minimize this expression (find the minimum time taken). Find dt Now soive dx = 0 to determine the value of x that minimizes the time taken. How do you know this is a valid value? For this value of x, find the minimum time possible and describe the route. Assumptions made in the problem The problem is perhaps more accurately stated as: You are standing at the edge of a river. You want to return to your campground which you can see further down the river on the other side. You must first swim across the river to any point on the opposite bank. From there you must run to the campground. What route will take the least amount of time? Look back at the original problem. What additional assumptions have been made in the original question? What information in the question are you unlikely to know when you are standing at the edge of the river? What additional information would you need to know to determine the shortest time possible? The original problem is a simplified version of a real-life situation. Criticize the original problem, and the information given, as much as possible. Extension In an exploration it is important to reflect critically on any made and the subsequent significance and limitations of Consider the open-box problem and the cylindrical can pr astions in Exercise 5R on page 267-268 these problems as