Question

The equation S = -16t2 + 34t + 184 models the height of a ball that is thrown upward from the roof of a 184 foot building and falls to the street below. In this equation S is the height in feet of the ball above the ground and t is the time in seconds the ball has traveled. According to this model, how many seconds did it take the ball to reach a height of 91 feet? (round to 1 decimal place)

Answer 1

t=1.6

Explanation:

• If the equation
  `S=-16t^2+34t+184` models the height of a ball above the ground, where t is time the ball travelled.
• If the height of the ball above the ground is S=91 instead of 184, then the time the ball should take to get to the ground comes from the expression above:
  `S=-16t^2+34t+91` (because now we want to know how much time does it takes to reach the ground if it is thrown from 91 foot, not 184).
• Then, to know when the ball reaches the floor, we must equal the equation to zero
  `-16t^2+34t+91=0` (because when the equation is zero, the height of the ball is zero, which means it is in the ground).

• To obtain the value of t in the expression
  `-16t^2+34t+91=0` , we can apply the well known formula
  `t=(-b(+-)√(b^2-4ac) )/(2a)`, where a is the coefficient that accompanies the quadratic term (in this case a=16), b is the coefficient that accompanies the linear term (b=34 in this case), and c is the constant coefficient (c=91).
• Because time is always possitive, we only retain the possitive value for t that solves the equation:
  `(-34(+-)√(34^2-4*(-16)*93) )/(2*16)`.
  `t=1.55\simeq1.6`