a baseball is hit with an initial velocity of 75 feet per second from the height of 3 feet.
The function h(t)=-16t squared + 75t + 3 models the path of the baseball where t represents the time in seconds that the ball has been traveling and h represents the height of the ball at time t.
1) after how many seconds is the height of the ball is 79.5 feet?
2) how long will it take for the ball to return to the ground?
3) how long will it take the ball to reach its maximum height? What is the mazimum height reached?
1. h = 16t^2 + 75t + 3 = 79.5 Ft.
-16t^2 + 75t -76.5 = 0.
Use Quadratjc Formula and get:
t = = 1.5 s.
2. d = Vo*t + 0.5g*t^2 = 90.9 Ft.
0 + 16*t^2 = 90.9.
t^2 = 90.9 / 16 = 5.68.
Tf = 2.38 s. = Fall time.
T = Tr + Tf = 2.34 + 2.38 = 4.72 s. =
Time in flight or time to return to gnd.
3a. Tr = (Vf-Vo)/g.
Tr = (0-75) / -32 = 2.34 s. = Rise time
3b. hmax=-16*(2.34)^2 + 75*2.34+3=90.9
Ft.
To find the answers to these questions, we need to solve the quadratic equation h(t) = -16t^2 + 75t + 3. Let's break down each question and solve them step by step:
1) To find after how many seconds the height of the ball is 79.5 feet, we need to set h(t) = 79.5 feet and solve the equation:
-16t^2+75t+3 = 79.5
Simplifying the equation, we get:
-16t^2 + 75t - 76.5 = 0
To find the value of t, we can use the quadratic formula:
t = [-b ± sqrt(b^2 - 4ac)] / 2a
For this equation, a = -16, b = 75, and c = -76.5. Plugging these values into the quadratic formula, we get:
t = [-75 ± sqrt(75^2 - 4(-16)(-76.5))] / (2*(-16))
Simplifying further will give us the two solutions for t, which represent the times when the ball reaches a height of 79.5 feet.
2) To find how long it will take for the ball to return to the ground, we need to find when the height of the ball is 0. So we set h(t) = 0 and solve the equation:
-16t^2 + 75t + 3 = 0
Using the quadratic formula as explained above, we find the values of t that correspond to when the ball hits the ground.
3) To find how long it will take for the ball to reach its maximum height, we need to find the vertex of the quadratic function h(t) = -16t^2 + 75t + 3. The x-coordinate of the vertex will give us the time, and the y-coordinate will give us the maximum height.
The time at the vertex can be found using the formula:
t = -b / 2a
For this equation, a = -16 and b = 75. Plugging in these values, we can calculate the time it takes for the ball to reach its maximum height. To find the maximum height reached, substitute this time into the equation h(t) and solve for h.
By following these steps and solving the equations accordingly, we can find the answers to the questions about the height and time of the baseball.