Warm Up Write down objective and homework Lay

Warm Up! • Write down objective and homework • Lay out homework (Area Worksheet) • Homework (Modeling Quadratic Worksheet) • Get a calculator!

Warm Up • Benchmark

• After the explosion in the oxygen tanks of the LEM (Lunar Excursion Module) on Apollo 13, mission control was given two options in aborting the mission. Option (1) was a direct abort in which the LEM would turn on its "retro-rockets" and head directly back home. Option (2) was to use the gravitational pull of the moon to slingshot the astronauts back to Earth. As history tells, option (2) was the option chosen. Option: 1 Option: 2

• Let's investigate the possibility of option (1) the potential of having the ship make an about -face and head away from the moon and back to Earth. Assume that Apollo 13 approaches the moon at a constant rate of speed, and then chooses to fire its "retro-rockets. " The spaceship slows down, and if all goes well, stops for an instant and then starts pulling away. While the rocket motor is firing, Apollo's distance from the surface of the moon can be described by a quadratic function.

Apollo 13 • Use your calculators to model this situation and analyze what would have happened to Apollo 13 had option (1) been chosen. Mission Control finds the following data for the location of Apollo 13

Apollo 13 • • • • After entering your data, make sure Plot 1 is turned on! Press Zoom Stat to see your data What do you see? This view may not give you enough of an idea of what the function looks like, so let's reset our window to show a broader view. Press WINDOW Set up your screen like this, using the up and down arrow keys to move around: Xmin = -5 Xmax = 30 Xscl = 5 Ymin = -60 Ymax = 600 Yscl = 100 Press GRAPH to see the new window you've set.

Apollo 13 • What does the X-axis represent (i. e. If you are sitting on the X-axis, where are you? )? • The surface of the moon! • Does it look like Apollo 13 will hit the moon or not? • It appears it’s likely to hit the moon • Now find the quadratic line of best fit • It’s the same way we find all lines of best fit, but we choose option 5 (quadreg)

Apollo 13 • What’s the equation? • y = 3 x 2 – 78 x + 500 • Now graph the equation (y=, VARS, Statistics, Right twice, Regeq) • Does your model tell you that Apollo 13 crashed into the surface of the moon, just touches the surface, or pulls away before reaching the surface? Explain. • Crashed into the surface because the regression has them going below the surface of the moon. • Approximately where is the rocket at t = 10 seconds, according to the model? • 20 feet from the moon • When would the rocket have crashed into the moon’s surface? • At 11. 47 seconds

Find the quadratic regression for the following

Answer the following • What’s the quadratic regression equation? • y = -0. 017 x 2 + 1. 93 x – 25. 34 • Determine the speed that maximizes miles per gallon. • 55. 38 miles per gallon • Predict the number of miles per gallon for a speed of 63 miles per hour. • 27. 21 miles per gallon • What speeds would result in using 23 miles per gallon? • 38. 08 mph and 72. 68 mph

You Try! • A. y = -16 x 2 + 33 x + 46 • B. 28. 5 feet

You Try! • What’s the quadratic regression equation? • Y = -0. 02 x 2 + 1. 15 x + 3. 73 • Determine the maximum amount of bushels per acre • 20. 17 acres • Determine how many pounds of fertilizer it took to get the most bushels per acre • 28. 67 pounds • Determine the fertilizer needed to have 8 bushels per acre • 4 pounds or 53. 3 pounds • Determine the bushels per acre if you have 50 pounds of fertilizer • 8 bushels per acre

You Try!

Answer the following • Determine the age at which an individual can expect to earn the most income. – 44 years old • Predict the peak income earned. – About $50, 173 • What does the model indicate as the median income of a newborn baby? – -$41, 021 • When does the model predict that people begin earning income? – Age 12 • At what age do people stop earning money? – Almost Age 78 • How valid are the results from questions 3 -6? Explain your reasoning.

Practice! • Are You Ready For Some Football? The height of a punted football can be modeled with the quadratic function h = -0. 01 x 2 + 1. 18 x + 2. The horizontal distance in feet from the point of impact with the kicker’s foot is x, and h is the height of the ball in feet. • What is the ball’s height when it has traveled 30 ft downfield – 28. 4 feet • What is the maximum height of the punt? How far downfield has the ball traveled when it reaches its maximum height? – Height: 36. 81 feet Time: 60 seconds • The nearest defensive player is 5 ft horizontally from the point of impact. How high must he get his hand to block the punt? – 7. 65 feet • Suppose the punt was not blocked but continued on its path. How far down field would the ball go before it hit the ground? – About 120 seconds • Why is the linear equation not a good model for the path of the football? Explain. – The football doesn’t keep a constant speed!

• More Football Although a football field appears to be flat, its surface is actually shaped like a parabola so that rain runs off to either side. The cross section of a field with synthetic turf can be modeled by y = -0. 000234(x – 80)2 + 1. 5, where x and y are measured in feet. • What is the field’s width? – About 160 feet • What is the maximum height of the field’s surface? – 1. 5 feet

GOOD PRACTICE! • http: //www 1. broward. edu/~hsorkin/Int. Web/Quadratic_Word_Problems. htm