IB Math Analysis & Approaches SL PDF Past Paper - Chapter 3

Chapter 3: Linear and Quadratic Functions 3.7 Applications of quadratics Essential Questions: When solvoing a real-life problem modeled by a quadratic equa- tion, how do you determine which method of solution to use? Investigation 1: Relief Supplies Crates of em...

Chapter 3: Linear and Quadratic Functions 3.7 Applications of quadratics Essential Questions: When solvoing a real-life problem modeled by a quadratic equa- tion, how do you determine which method of solution to use? Investigation 1: Relief Supplies Crates of emergency supplies are sometimes dropped from cargo planes to humanitarian aid workers. Each crate has its own parachute. Pilots must determine the height and time of the drop that will allow the containers to reach the ground intact. The crate is said to be in free fall during the time before the parachute opens. The functions h and g give the crate’s height above the ground, measured in metres, t seconds after the crate leaves the plane. During free fall: h(t) = −4.9t 2 + 720 With parachute open: g (t) = −5t + 670 Why are two models needed to represent the height of the crate? IB Math Analysis & Approaches SL 1/12 Investigation 1: Relief Supplies The functions h and g give the crate’s height above the gorund, measured in metres, t seconds after the crate leaves the plane. During free fall: h(t) = −4.9t 2 + 720 With parachute open: g (t) = −5t + 670 1. For the model representing the motion during free fall, what does the constant value 720 represent? 2. For the model representing motion with the parachute open, what does −5, the coefficient of t, represent? IB Math Analysis & Approaches SL 2/12 Investigation 1: Relief Supplies The functions h and g give the crate’s height above the ground, measured in metres, t seconds after the crate leaves the plane. During free fall: h(t) = −4.9t 2 + 720 With parachute open: g (t) = −5t + 670 3. How could you transform the parent graph h(t) = t 2 to give the function h(t) = −4.9t 2 + 720? What do these transformations tell you about the motion of the crate in this context? 4. Without using your calculator, sketch a graph of this piecewise function and label key features. IB Math Analysis & Approaches SL 3/12 Investigation 1: Relief Supplies The functions h and g give the crate’s height above the ground, measured in metres, t seconds after the crate leaves the plane. During free fall: h(t) = −4.9t 2 + 720 With parachute open: g (t) = −5t + 670 5. How long after the crate leaves the plane does the parachute open? 6. How long after the crate leaves the plane does it reach the ground? 7. What is the domain of h and g in the context of this problem? IB Math Analysis & Approaches SL 4/12 Investigation 2: Suspension Bridge The main span of the Clifton Suspension Bridge in Bristol, UK is about 194 m long. The height of the towers is 26.2 m. The supporting cable for the main span of the bridge touches the road deck at the centre of the span. 1. The parabola shown in the graph models the supporting cable of the main span of the Clifton Suspension Bridge and the green line segments represent the towers. Write down the values of r , s, and t. IB Math Analysis & Approaches SL 5/12 Investigation 2: Suspension Bridge 2. Consider the three forms of the equation of a quadratic function. For each of these forms, the values of some of the parameters are equal to r , s, or t. y = a(x − h)2 + k y = ax 2 + bx + c y = a(x − p)(x − q) Replace the appropriate parameters in each equation with the values of r , s, and t. IB Math Analysis & Approaches SL 6/12 Investigation 2: Suspension Bridge 3. Choose one of the equations from question 2 and find any other missing parameters to complete a quadratic function that models the supporting cable. 4. Which form of the quadratic function did you use and why? 5. Define the variables in your function. 6. How can the quadratic function help you answer questions about the supporting cable of the Clifton Suspension Bridge? IB Math Analysis & Approaches SL 7/12 Investigation 2: Suspension Bridge 7. Find the height of the supporting cable from the road deck at a point 24 m from a tower. 8. Suppose there are vertical cables with length 2.75 m, attaching the supporting cable to the road deck. Find the distance of these cables from the left-hand tower. IB Math Analysis & Approaches SL 8/12 Quadratic Models A rancher plans to use 120 m of fencing to build a rectangular pen. 1. Let x represent the width of the pen. Find the length and area of the pen in terms of x. 2. Find the dimensions of the pen if the area is 800 m2. 3. Find the maximum possible area of the pen. IB Math Analysis & Approaches SL 9/12 Quadratic Models A bakery sells apple pies for $12 each, and sells an average of 40 apple pies per day. The owner estimates that for each $0.50 increase in price, the average sales will decrease by one pie a day. Use tables, graphs, and/or equations to model the problem. Use your model to determine what sale price the bakery owner should charge to maximize the revenue from apple pie sales. IB Math Analysis & Approaches SL 10/12 Textbook Exercises Textbook Exercises To practice and apply these concepts, work through the following exercises: Exercise 3X p.176 #1-5 IB Math Analysis & Approaches SL 11/12  Assignment: 1. Process your notes Chunk the information by numbering each big idea Circle key words and new vocabulary Highlight main ideas Fill in any gaps and add clarifying comments Write questions next to any points of confusion 2. Textbook Exercises: Read pp.173-176 3X p.176 #2, 3, 5 IB Math Analysis & Approaches SL 12/12