A ball is thrown in an upward direction off a platform that is 45 feet high with an initial velocity of 60 feet per second. The height, in feet, of the ball at the time t is given by
h(t)= -16t^2 + 60t + 45. The time, t, is given in seconds. Round all answers to the hundredths.
1. At what time, after the start, is the ball 45 feet high again?
2. Find the time when the ball hits the ground.
1.
45 = 45 +60 t -16 t^2
4 t( 15-4t) = 0
t = 0 of course and t =15/4
2.
0 = 45 + 60 t -16 t^2
solve quadratic a = -16, b = 60 , c = 45
https://www.mathsisfun.com/quadratic-equation-solver.html
To find the time when the ball is 45 feet high again, we need to solve the equation h(t) = 45.
1. At what time is the ball 45 feet high again?
To find the time, t, we can substitute h(t) = 45 into the equation and solve for t.
-16t^2 + 60t + 45 = 45
Simplifying the equation:
-16t^2 + 60t = 0
Dividing both sides by 5 to simplify:
-3t^2 + 12t = 0
Factoring out t:
t(-3t + 12) = 0
From this equation, we have two possibilities:
a) t = 0
b) -3t + 12 = 0
From option a, we can see that t = 0 is the initial time, when the ball is thrown. We need to find the time after the start, so the only option is option b.
-3t + 12 = 0
Simplifying:
-3t = -12
t = -12/-3
t = 4
Therefore, the ball will be 45 feet high again after 4 seconds.
2. Find the time when the ball hits the ground.
To find the time when the ball hits the ground, we need to find the time, t, when h(t) = 0.
-16t^2 + 60t + 45 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 60, and c = 45. Substituting these values into the quadratic formula:
t = (-60 ± √(60^2 - 4(-16)(45))) / (2(-16))
Simplifying:
t = (-60 ± √(3600 + 2880)) / (-32)
t = (-60 ± √6480) / (-32)
Since we are looking for the time when the ball hits the ground, we only consider the positive value of t, as time cannot be negative. Therefore:
t = (-60 + √6480) / (-32)
Calculating this value, we find:
t ≈ 3.51
Therefore, the ball hits the ground approximately 3.51 seconds after it is thrown.