Quadratics in Real Life

Years after 1940 10 20 30 40 50 60 Avg. No. Of Miles per Gallon 14.8 13.9 13.4 13.5 15.9 20.1 22.0 Use your graphing calculator to create a scatter plot of the data. Explain in DETAIL why a quadratic function is appropriate for modeling this. Create a regression from the data. What is the quadratic equation you found? Use the quadratic model you found in b. to determine the worst year for automobile fuel efficiency. What was the average number of miles per gallon for that year? 1. The following data shows fuel efficiency, in miles per gallon, for all U.S. automobiles in the indicated year. 𝟎.𝟎𝟎𝟓𝟎𝟏𝟐 𝒙 𝟐 −𝟎.𝟏𝟕𝒙+𝟏𝟒.𝟖𝟐 Find the vertex 𝟏𝟔.𝟔𝟕, 𝟏𝟑.𝟑𝟖 which means in 1956 about 13.38 mpg. Use the quadratic model from part b. predict the avg number of miles per gallon for a US Automobile in 2010. Does this make sense? 2010 is 70 years after 1940, so 𝟎.𝟎𝟎𝟓𝟎𝟏𝟐 𝟕𝟎 𝟐 −𝟎.𝟏𝟕 𝟕𝟎 +𝟏𝟒.𝟖𝟐≈𝟐𝟕 mpg

2. ODU students drop pumpkins out of windows and records the data. Time t (sec) 0.5 1 1.5 2 2.5 3 Distance Fallen 𝑑= 16𝑥 2 Height Above Ground ℎ=100−𝑑 100 𝟒 𝟏𝟔 36 64 𝟏𝟎𝟎 𝟏𝟒𝟒 96 84 64 36 -44 Use the function 𝑑= 16𝑥 2 (ignoring resistance) to complete table. Use data found, to create scatter plot only using TIME and HEIGHT ABOVE GROUND. Create a regression. What is the equation you found? What solution does the equation produce at 4 seconds? 2. ODU students drop pumpkins out of windows and records the data. 𝒚=−𝟏𝟔 𝒙 𝟐 +𝟏𝟎𝟎 𝒚=−𝟏𝟔 𝟒 𝟐 +𝟏𝟎𝟎=−𝟏𝟓𝟔 feet Does this make sense? Yes, because it hits the ground at 2.5 seconds, so anything after would be negative, or “below” ground.

3. Now we are launching pumpkins with a cannon! Suppose compressed-air cannon fires a pumpkin straight up into the air from a height of 20 feet and provides an initial upward velocity of 90 feet per second. Use the function 𝑓 𝑡 = ℎ 𝑜 + 𝑣 𝑜 𝑡− 16𝑡 2 (where 𝑓 𝑡 is a function of height of the object with respect to time) to create an appropriate function rule that would combine these conditions with the effect of gravity. What is the maximum height? How long did it take to get there? How would you change your function rule in Part a if the pumpkin is launched at a height of 15 feet with an initial upward velocity of 120 feet per second? 3. Now we are launching pumpkins with a cannon! 𝒇 𝒕 =𝟐𝟎+𝟗𝟎𝒕−𝟏𝟔 𝒙 𝟐 Find the vertex (2.81 sec, 147 feet) 𝒇 𝒕 =𝟏𝟓+𝟏𝟐𝟎𝒕−𝟏𝟔 𝒙 𝟐 What is the maximum height? How long did it take to get there? Find the vertex (3.75 sec, 240 feet)

𝒉=𝟑+𝟑𝟔𝒕− 𝟏𝟔𝒕 𝟐 𝒊𝒏𝒊𝒕𝒊𝒂𝒍 𝒉𝒆𝒊𝒈𝒉𝒕 𝒊𝒔 𝟓 𝒇𝒆𝒆𝒕 4. Billie throws a ball to second base during baseball practice. The equation resulting from his throw is represented as ℎ=5+42𝑡− 16𝑡 2 . Determine th initial height of the ball and the initial velocity when thrown. What is the maximum height? How long did it take to get there? 𝒊𝒏𝒊𝒕𝒊𝒂𝒍 𝒉𝒆𝒊𝒈𝒉𝒕 𝒊𝒔 𝟓 𝒇𝒆𝒆𝒕 Initial velocity is 42 feet per second Find the vertex (1.31 sec, 32.6 feet) 5. A soccer ball is kicked at a height of 3 feet and the ball had an initial velocity of 36 f/s. Write an equation that represents the height of the soccer ball (h) at any time (t) during the flight. What is the maximum height? How long did it take to get there? 𝒉=𝟑+𝟑𝟔𝒕− 𝟏𝟔𝒕 𝟐 Initial velocity is 42 feet per second Find the vertex (1.12 sec, 23.3 feet)

6. The graph represents how the height of a field goal attempt is related to the horizontal distance of the field goal attempt… From what height was the football kicked? Does this make sense? Why? If the crossbar of the goalpost is 10 ft. above the ground and is 90 ft. away from where the ball is kicked, will the ball have enough height when it reaches the goalpost to go over the crossbar? Explain. It appeas the football was kicked at 0 feet high, which means it was on the round to start. Yes this makes sense, because a holder will receive the snap and hold it in place for the kicker. And the football is held on the ground. It should reach the cross bar, but might not go through. It might hit the bottom of the crossbar and bounce off.

Find the vertex (1.31 sec, 32.6 feet) 7. Katie, a goalie for Washing Town High School’s soccer team, needs to get the ball downfield to her teammates on the offensive end of the field. She punts the ball from a point 2.5 feet above the ground with an initial upward velocity of 40 feet per second Write a function rule that relates the ball’s height above the field h to its time in the air t. Use this rule to estimate the time when the ball will hit the ground. (HINT: h=0) Suppose Katie were to kick the ball right off the ground with the same initial upward velocity. Do you think the ball would be in the air the same amount of time, for more time, or for less time? Check your thinking. 𝒉 𝒕 =𝟐.𝟓+𝟒𝟎𝒕−𝟏𝟔 𝒕 𝟐 Find the vertex (1.31 sec, 32.6 feet) 𝒉 𝒕 =𝟐.𝟓+𝟒𝟎𝒕−𝟏𝟔 𝒕 𝟐 =𝟎 So find, x-intercepts (-0.061, 0) and (2.56, 0) are the x-intercepts Since you cannot have negative amounts of time, the time would be 2.56 seconds. 𝒉 𝒕 =𝟒𝟎𝒕−𝟏𝟔 𝒕 𝟐 (0, 0) and (2.5, 0) are the x-intercepts Less time, 2.5 seconds compared to 2.56 seconds.

Find the vertex (1.31 sec, 32.6 feet) 8. A basketball player takes a shot. The graph shows the height of the ball, starting when it leaves the player’s hands. Estimate the height of the ball when the player releases it. When does the ball reach its maximum height? What is the maximum height? How long does it take the ball to reach the basket (a height of 10 feet)? About 6.5 feet Find the vertex (1.31 sec, 32.6 feet) Appears to be about 17 feet and that takes about 0.8 seconds Appears to be about 1.5 seconds

At about 2 seconds the rocket is 148 feet above the ground. 9. A model rocket is launched from the top of a hill. The table shows how the rocket’s height above ground level changes as it travels through the air. How high above ground level does the rocket travel? When does it reach this maximum height? From what height is the rocket launched? How long does it take the rocket to return to the top of the hill? 𝒉 𝒕 =𝟖𝟒+𝟔𝟒𝒕−𝟏𝟔 𝒕 𝟐 At about 2 seconds the rocket is 148 feet above the ground. 𝒊𝒏𝒊𝒕𝒊𝒂𝒍 𝒉𝒆𝒊𝒈𝒉𝒕 𝒊𝒔 𝟖𝟒 𝒇𝒆𝒆𝒕 About 4 seconds

𝑫 𝟏𝟎 =−𝟒 𝟏𝟎 𝟐 +𝟏𝟐 𝟏𝟎 =−𝟐𝟖𝟎𝒇𝒆𝒆𝒕 𝑫 𝟖.𝟓 =−𝟒 𝟖.𝟓 𝟐 +𝟏𝟐 𝟖.𝟓 =−𝟏𝟖𝟕𝒇𝒆𝒆𝒕 10. A ship drops an anchor offshore Honiara. When the anchor is tossed into the water, the depth in feet D it has descended after t seconds is given by the equation: 𝑫 𝒕 =−𝟒 𝒕 𝟐 +𝟏𝟐𝒕 If it takes the anchor 10 seconds to reach the bottom, how deep is the water where the ship has dropped anchor? If the ship moves to another location and the anchor takes 8.5 seconds to reach the bottom, how deep is the water in that spot? If the ship anchors in the Harbor of Honiara, where the water is 72 feet deep, how long will it take for the anchor to reach the bottom when it is dropped? What is the maximum depth the anchor can reach? How long will it take to reach the maximum depth? 𝑫 𝟏𝟎 =−𝟒 𝟏𝟎 𝟐 +𝟏𝟐 𝟏𝟎 =−𝟐𝟖𝟎𝒇𝒆𝒆𝒕 𝑫 𝟖.𝟓 =−𝟒 𝟖.𝟓 𝟐 +𝟏𝟐 𝟖.𝟓 =−𝟏𝟖𝟕𝒇𝒆𝒆𝒕 𝑫 𝒕 =−𝟒 𝒕 𝟐 +𝟏𝟐𝒕=−𝟕𝟐𝒇𝒆𝒆𝒕 →−𝟒 𝒕 𝟐 +𝟏𝟐𝒕+𝟕𝟐=𝟎 (-3, 0) and (6, 0) Can’t have negative time, so it will be 6 seconds Find the vertex (1.5 sec, 9 feet)

11. Metropolitan Container produces storage containers from recycled plastic. The total cost in dollars C of manufacturing n containers is given by the equation: 𝑪 𝒏 =𝟐 𝒏 𝟐 +𝟗𝒏+𝟏𝟎𝟎 What is the total cost of manufacturing 4 containers? What is the cost of manufacturing 10 containers? What is the minimum cost? How many containers will be produced? Make sense? 𝑪 𝟒 =𝟐 𝟒 𝟐 +𝟗 𝟒 +𝟏𝟎𝟎=$𝟏𝟔𝟖 𝑪 𝟏𝟎 =𝟐 𝟏𝟎 𝟐 +𝟗 𝟏𝟎 +𝟏𝟎𝟎=$𝟑𝟗𝟎 Find the vertex (-2.25, 89.9)