Lesson 9

Quadratic equations can be used to model situations in many contexts. You have looked at trajectories of projectiles and business situations. You will now have a chance to hone your modelling and solving skills by studying an example. Then you get to try a few problems yourself.

 

Example 1: Stream of Water

 

It is a hot day and you and a friend are at a park with a cooling fountain. The stream of water leaves the nozzle 10 cm above the water’s surface. The trajectory of the water stream reaches a height of 200 cm above the water when it is 300 cm from the nozzle. Your friend reaches out to touch the stream. What is the height of the stream of water when it is a horizontal distance of 70 cm from the nozzle, which is as far as she can safely reach?

 

To answer the question, place the origin at the highest point. Draw a sketch of the situation that includes all of the given measurements.

 

 

Assume that the trajectory of the water in the stream is a perfect parabola. Also assume that all measurements are to the nearest centimetre and that the water is perfectly level.

 

With the origin at the vertex of the parabola, the y-intercept is at (0, 0), so the quadratic function y = ax2 + bx + c reduces to y = ax2.

 

To solve for a, take the coordinates of the nozzle, (−300, −190) measured in centimetres, and insert the coordinates into the function.

 

 

The function can be written as or as the approximation −0.002 11x2.

 

The horizontal distance of 70 cm from the nozzle, as far as your friend can safely reach, will be −300 + 70, or −230 cm, from the vertex. You want the height of the stream of water at that point, which is related to the value of y. So, substitute −230 in for x and solve for y.

 

 

The stream of water will be about 112 cm below the vertex. That means the stream will be 200 − 112, or 88 cm, above the water at a horizontal distance of 70 cm from the nozzle.

 

Self-Check 1

 

Suppose the tip of the nozzle had been chosen as the point of origin in Example 1. The sketch would be a similar shape. However, the vertex would be at (300, 190) and the axes would be in a different place.

1. Draw a sketch of the graph with the origin at the tip of the nozzle. Answer
   
   
2. Write a quadratic function in the vertex form to model the situation, and calculate the value of the coefficient a. Why would the vertex form be an excellent choice here? Answer
   
   
3. What is the height of the stream of water when the stream is a horizontal distance of 70 cm from the nozzle? Answer