Applications Involving Quadratic Equations

Question

A child throws a water balloon off the roof of a building. The height h (in feet) of the balloon, t seconds after being launched, is given by the function h(t) = -16t² + 48t² + 64.

 

a. Find any values of t for which the height of the balloon is 100 feet.

 

b. Find any values of t for which the height of the balloon is 80 feet. (Give decimal approximations accurate the nearest thousandth.)

Answer 1

First, I think there is a typo: the 48t² should be 48t  .

The balloon is starting at 64 feet high (when t=0)

and the initial velocity is 48 ft/sec going up;

gravity acts through the -16t² term.

 

a. We want:   -16t² + 48t + 64 = 100

        -16t² + 48 - 36 = 0

   Divide all terms by -4 to simplify somewhat:

        4t² - 12t + 9 = 0

  Could use the quadratic formula, but it factors nicely, so:

       (2t - 3) (2t - 3) = 0

  Two identical solutions:  

       2t - 3 = 0   gives  

          t = 3/2 seconds   (or 1.5 s  as decimal)

  Because the two solutions are identical this is the height at the peak of the balloon's trajectory.

 

b. Now we want: -16t² + 48t + 64 = 80

   (Note, this should happen once before the 1.5 s peak and once after.)

    As before we subtract the 80 to get 0

        -16t² + 48t - 16 = 0

   all terms can be divided by -16 to get:

      t² -3t + 1 = 0

   Now use the quadratic formula:

   t =  "-b +/- sqrt(b² - 4ac)   all over 2a"

   t = -(-3) +/- sqrt( (-3)² - 4 * 1 * 1 )    all over  2

 

     = 1.5  +/- 0.5*sqrt(9-4)

     = 1.5 +/- 0.5 * sqrt(5)

     = 1.5 +/- 1.1180

     =  2.618 and 0.382  seconds.

 

Check these by plugging each one back into the original equation to get 80.