susan throws a softball upward into the air at a speed of 32 feet per second from a 8 foot platform. the distance upward that the ball travels is given by the function. d(t)=-16t^2+32t+8. What is the maximum height of the softball? how many seconds does it take to reach the ground after first being thrown.
To find the maximum height of the softball, we need to determine the vertex of the quadratic function d(t) = -16t^2 + 32t + 8. The vertex represents the highest point or the maximum height.
The vertex of the quadratic function with the general form f(t) = ax^2 + bx + c can be found using the formula t = -b/(2a). In this case, the equation d(t) = -16t^2 + 32t + 8 corresponds to f(t) = -16t^2 + 32t + 8, where a = -16, b = 32, and c = 8.
Calculating t = -b/(2a) = -32/(2*(-16)), we get t = -32/(-32) = 1.
So, the maximum height of the softball occurs at t = 1 second. To find the maximum height, we substitute t = 1 into the equation d(t):
d(1) = -16(1)^2 + 32(1) + 8
= -16 + 32 + 8
= 24
Therefore, the maximum height of the softball is 24 feet.
To determine how many seconds it takes for the softball to reach the ground after being thrown, we need to find the time when d(t) = 0. This represents the moment the ball hits the ground.
We can solve the quadratic equation -16t^2 + 32t + 8 = 0 by factoring, completing the square, or using the quadratic formula. For simplicity, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
Plugging in a = -16, b = 32, and c = 8 into the formula, we get:
t = (-32 ± √(32^2 - 4*(-16)*8))/(2*(-16))
t = (-32 ± √(1024 + 512))/(-32)
t = (-32 ± √1536)/(-32)
t = (-32 ± 39.2)/(-32)
Using the positive root:
t ≈ -32 + 39.2)/(-32)
t ≈ 7.2/(-32)
t ≈ -0.225
As time cannot be negative in this context, the ball takes approximately 0.225 seconds to reach the ground after being thrown.
Therefore, it takes about 0.225 seconds for the softball to reach the ground after first being thrown.
as you know, the vertex of a parabola occurs at x = -b/2a.
Here, that is -32/-32 = 1
d(1) = 24
solving d=0 for t, we have t = 1+√3/2