a punter kicked the football into the air with an upward velocity of 62 ft/s. its height in feet after t seconds is given by the function h = -16t^2 + 62t +2. What is the maximum height the ball reaches? How long will it take the football to reach the maximum height? How long does it take for the ball to hit the ground?
b. T = Tr + Tf = 1.94 + 1.94 = 3.88 s.
Tr = Rise time.
Tf = Fall time.
The answer is B. 1.94 s, 62.06
To find the maximum height the ball reaches, we need to determine the vertex of the given function. The vertex is the highest point on the graph and represents the maximum height. The vertex of a quadratic function in the form h = ax^2 + bx + c can be found using the formula -b/2a.
In this case, the function represents the height of the ball (h) as a function of time (t), given by h = -16t^2 + 62t + 2. Comparing it to the standard form, we can see that a = -16 and b = 62. Plugging these values into the formula, we get:
t = -62 / (2 * -16) = 1.9375 seconds
Therefore, the maximum height the ball reaches is given by plugging t = 1.9375 back into the equation:
h = -16(1.9375)^2 + 62(1.9375) + 2 ≈ 60.96 feet
So, the maximum height the ball reaches is approximately 60.96 feet.
To find the time it takes for the football to reach the maximum height, we use the value of t we found earlier:
t = 1.9375 seconds
Therefore, it takes the football approximately 1.9375 seconds to reach its maximum height.
To find the time it takes for the ball to hit the ground, we need to find the value of t when the height (h) is equal to zero. We can set the equation h = 0 and solve for t:
-16t^2 + 62t + 2 = 0
This quadratic equation can be factored or solved using the quadratic formula. Solving it using the quadratic formula, we get:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values a = -16, b = 62, and c = 2 into the formula, we get:
t = (-62 ± √(62^2 - 4 * -16 * 2)) / (2 * -16)
Simplifying the equation further, we have:
t = (-62 ± √(3844 + 128)) / (-32)
t = (-62 ± √3972) / -32
Taking the positive square root (since time cannot be negative), we have:
t ≈ (-62 + √3972) / -32 ≈ 4.4381 seconds
Therefore, it takes approximately 4.4381 seconds for the ball to hit the ground.
t = -B/2A = -62/-32 = 1.94
h = -16*1.94^2 + 62*1.94 + 2 = 62.1 Ft.