Objective: Students will be able to (SWBAT) solve quadratic equations with real coefficient that have complex solutions, in order to (IOT) make sense of a real life situation and interpret the results in the context of the problem. Standards: MGSE9-12.N.CN.7 Solve Quadratics. Solve quadratic equations with real coefficients that have complex solutions by (but not limited to) square roots, completing the square, and the quadratic formula.
Bell Ringer: When a bus turns a corner, it must swing out so that its rear wheels don t go into the cycle lane. In this picture, as the bus goes round the corner, the front wheel is on the edge of the cycle lane, but the rear wheel cuts into the cycle lane.
The diagram on the right shows the geometry of this situation.
The distance between the front and rear wheel is called the wheelbase,!.!
Let! represent the radius of the outside edge of the cycle lane. "!
The distance marked! shows the amount by which the rear wheels cuts into the cycle lane. "! #
Having difficulty getting started: Describe fully the position in the road of the front and back wheels at the start of the turn. Describe fully the position in the road of the front and back wheels at the end of the turn. What do you know? What do you need to find out? What shape can you see in the diagram? What mathematics have you studied that connects with the situation? What is the connection between! and "?
1. Use the diagram to show that! " 2!& + ( " = 0. The radius of the circle formed by the back wheel is +,. This radius is perpendicular to the tangent to the circle, the line segment formed by the wheelbase. Thus &, &! and ( form a rightangled triangle., +, - +
1. Use the diagram to show that! " 2!& + ( " = 0. Using the Pythagorean Theorem: - +, -,
2. Let w = 10 feet and r = 17 feet. (a) Figure out how much the rear wheel cuts into the cycle lane. If we take w = 10 and r = 17 and substitute into the equation:! " 34! + 100 = 0 There is a range of methods we might use to figure out the value for!.
Completing the square:! " 34! = 100! " 34! + 289 = 100 + 289 (! 17) " = 189 Taking the square root of both sides.! 17 = 13.75 or! 17 = 13.75! = 30.75 or! = 3.25. The outside edge of the cycle lane is a circle of radius only 17 feet, therefore! = 30.75 feet is not a possible solution in this context. The bus cuts into the cycle lane by 3.25 feet (3 feet 3 inches).
Quadratic Formula: The outside edge of the cycle lane is a circle of radius only 17 feet, therefore! = 30.75 feet is not a possible solution in this context. The bus cuts into the cycle lane by 3.25 feet (3 feet 3 inches).
Replacing! # with the variable $. % & = ( & + * & 17 & = ( & + 10 & ( & = 289 100 = 189 ( = 13.75 or ( = 13.75. 4 = % ( 4 = 17 13.75 or 4 = 17 + 13.75 4 = 3.25 or 4 = 30.75 The outside edge of the cycle lane is a circle of radius only 17 feet, therefore 4 = 30.75 feet is not a possible solution in this context. The bus cuts into the cycle lane by 3.25 feet (3 feet 3 inches).
(b) Figure out how far the front wheel must be from the outside edge of the cycle lane for the rear wheel not to cut into the cycle lane. We need to reinterpret the situation, to find the value of! when " = 0. Using the Pythagorean Theorem:
A negative value for b is impossible given the context, therefore the front wheel is: 19.72 17 = 2.72 feet (approx. 2 feet 9 inches) from the outside edge of the cycle lane.
We can also solve the quadratic equation: Solve this equation by either completing the square or using the quadratic formula.
1. Take turns to work through a student s solution. Try to understand the method the student has chosen and any errors he or she has made. 2. Explain your answer to the rest of the group. 3. Listen carefully to explanations. Ask questions if you don t understand. 4. Once everyone is satisfied with the explanations, write the answers to the questions below the student s solution. Make sure the student who writes the answers is not the student who explained them.
Sample Response to Discuss: Guy! 2%& 34++/5"&+!"#$%&'()' (%*&$#+'+#,+' ()'-.-/+'/"0+1 Where has Guy made a mistake? What does he need to do to correct this mistake?
Sample Response to Discuss: Ryan 6$#+7"/8! 2%& 34++/5"&+!"#$%&'()' (%*&$#+'+#,+' ()'-.-/+'/"0+1!("# Does Ryan's result make practical sense? Why/Why not?
Sample Response to Discuss: Donna 6$#+7"/8! 2%& 34++/5"&+!"#$%&'()' (%*&$#+'+#,+' ()'-.-/+'/"0+1!("# Does Donna's result make practical sense? Why/Why not? Sketch the graph from! = # to! = $#. What value of! should Donna try?
Which of the student s methods did you think was the most effective? Why? Which approach did you find difficult to understand? Why?
Closing Question: Liam Check Liam s work carefully and correct any errors you find. 6$#+7"/8! 2%& 34++/5"&+!"#$%&'()' (%*&$#+'+#,+' ()'-.-/+'/"0+1!("# What method has Liam used? Do you think Liam has chosen a good method? Explain your answer. What isn t clear about the work? Why is Liam s work incomplete?
Closing Question: Liam Liam has used an efficient method, however he has not labeled the diagram. Liam should explain that a represents the radius of the circle turned by the rear wheel. 2%&! 34++/5"&+ 6$#+7"/8!("#!"#$%&'()' (%*&$#+'+#,+' ()'-.-/+'/"0+1
Closing Question: Liam Liam makes a mistake in his manipulation of the equation. The solution should be:! 2 = 289 100 = 189! = 13.75 or! = 13.75. A negative value for! is not possible in this context, therefore - = 17 13.75 = 3.25. The bus cuts into the cycle lane by 3.25 feet (3 feet 3 inches).
7. A cliff diver on a Caribbean island jumps from a height of 105 feet, with an initial upward velocity of 5 feet per second. An equation that models the height, h %, above the water, in feet, of the diver in time elapsed, t, in seconds, is h % = 16% 2 + 5% + 105. How many seconds, to the nearest hundredth, does it take the diver to fall 45 feet below his starting point?
8. A homeowner wants to increase the size of a rectangular deck that now measures 14 feet by 22 feet. The building code allows for a deck to have a maximum area of 800 square feet. If the length and width are increased by the same number of feet, find the maximum number of whole feet each dimension can be increased and not exceed the building code.