Casey is a t bat and hits a home run ball the height of the ball in feet can be modeled by the function h(t)=-9t^2+72t where t is time in seconds.
But wait! Jim the center fielder leaps into the air according to the functiong(t)=-15t^2+240t-951
1)Find the intersection?
2)did Jim time his jump right to catch the ball?
3)if the fence is 8ft high did Jim jump high enough to catch the ball and prevent the home run?
4)what assumptions are we making if we conclude that Jim caught the ball?
(1) Find the intersection of `f(t)=-9t^2+72t` and `g(t)=-15t^2+240t-951`
The intersection(s) are at the point(s) where `f(t)=g(t)` :
Using the quadratic formula we get:
`t=14+-(5sqrt(6))/2~~7.876 "or" 20.124` . At t=20.124 the height is negative so we use t=7.88 seconds.
(2) Jim and the ball are at the same height at `t~~7.88` seconds, so yes.
(3) The height of the ball and Jim at `t~~7.88` seconds is approximately 8.51 feet (`-9(7.88^2)+72(7.88)` ) so they are over the fence, so yes he is high enough.
(4) We assume that the ball travelled far enough horizontally, and that Jim is in line with the path of the ball and that the time `t` is the same for both the ball and Jim.