Problem 2 Using the Vertical Motion Model

Entertainment During halftime of a basketball game, a sling shot launches T-shirts at the crowd. A T-shirt is launched with an initial upward velocity of 72 ft/s. The T-shirt is caught 35 ft above the court. How long will it take the T-shirt to reach its maximum height? What is its maximum height? What is the range of the function that models the height of the T-shirt over time?

The function h = − 16 t 2 + 72 t + 5 h equals negative , 16 t squared , plus 72 t plus 5  gives the T-shirt's height h, in feet, after t seconds. Since the coefficient of t 2 t squared  is negative, the parabola opens downward, and the vertex is the maximum point.

• Method 1 Use a formula.

  t = − b 2 a = − 72 2 ( − 16 ) = 2.25 Find the  t -coordinate of the vertex . h = − 16 ( 2.25 ) 2 + 72 ( 2.25 ) + 5 = 86 Find the  h -coordinate of the vertex . table with 2 rows and 3 columns , row1 column 1 , t , column 2 equals . fraction negative b , over 2 eh end fraction . equals . fraction negative 72 , over 2 . open , negative 16 , close end fraction . equals , 2.25 , column 3 cap findthe . t . minuscoordinateofthevertex . . , row2 column 1 , h , column 2 equals negative 16 . open , 2.25 , close squared . plus 72 . open , 2.25 , close . plus 5 equals 86 , column 3 cap findthe . h . minuscoordinateofthevertex . . , end table

  The T-shirt will reach its maximum height of 86 ft after 2.25 s. The range describes the height of the T-shirt during its flight. The T-shirt starts at 5 ft, peaks at 86 ft, and then is caught at 35 ft. The height of the T-shirt at any time is between 5 ft and 86 ft, inclusive, so the range is 5 ≤ h ≤ 86 . 5 less than or equal to h less than or equal to 86 .

• Method 2 Use a graphing calculator.

  Enter the function h = − 16 t 2 + 72 t + 5 h equals negative , 16 t squared , plus 72 t plus 5  as y = − 16 x 2 + 72 x + 5 y equals negative , 16 x squared , plus 72 x plus 5  on the Y = cap y equals screen and graph the function.

  

  Use the CALC feature and select MAXIMUM. Set left and right bounds on the maximum point and calculate the point's coordinates. The coordinates of the maximum point are (2.25, 86).

  The T-shirt will reach its maximum height of 86 ft after 2.25 s. The range of the function is 5 ≤ h ≤ 86 . 5 less than or equal to h less than or equal to 86 .