A ball is thrown directly upward from an initial height of 200 feet with an initial velocity of 96 feet per second. After t seconds, the height of the ball above the ground is s(t) = 16t2+ 96t + 200.
a. After how many seconds will the ball reach its maximum height?
b. What was the maximum height?
c. How long will it take before the ball reaches the ground?
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To find the answers to these questions, we can use the formula for the time it takes for a projectile to reach its maximum height, the formula for the maximum height, and the quadratic formula to find when the ball reaches the ground.
a. To find the time it takes for the ball to reach its maximum height, we can use the formula for the time of flight of a projectile, which is given by the equation:
t_max = -b / (2a)
where a = -16 and b = 96.
Substituting these values into the equation, we get:
t_max = -96 / (2*(-16))
t_max = -96 / (-32)
t_max = 3
Therefore, the ball will reach its maximum height after 3 seconds.
b. To find the maximum height, we can substitute the value of t_max (3 seconds) into the equation for the height:
s(t) = -16t^2 + 96t + 200
s(3) = -16(3)^2 + 96(3) + 200
s(3) = -16(9) + 288 + 200
s(3) = -144 + 288 + 200
s(3) = 344
Therefore, the maximum height of the ball is 344 feet.
c. To find when the ball reaches the ground, we can set the height equal to 0 and solve for t using the quadratic formula:
-16t^2 + 96t + 200 = 0
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
where a = -16, b = 96, and c = 200.
t = (-96 ± √(96^2 - 4*(-16)*200)) / (2*(-16))
t = (-96 ± √(9216 + 12800)) / (-32)
t = (-96 ± √(22016)) / (-32)
Since time cannot be negative in this context, we can ignore the negative value of t.
t = (-96 + √(22016)) / (-32)
t = (-96 + 148.37) / (-32)
t ≈ -52.37 / -32
t ≈ 1.64
Therefore, it will take approximately 1.64 seconds for the ball to reach the ground.
To find the maximum height of the ball and the time it takes to reach it, we need to find the vertex of the quadratic function s(t) = -16t^2 + 96t + 200, which represents the height of the ball as a function of time.
a. The ball reaches its maximum height at the vertex of the parabolic curve. The x-coordinate of the vertex can be found using the formula t = -b / (2a), where a is the coefficient of t^2 and b is the coefficient of t in the quadratic equation.
In this case, a = -16 and b = 96. Plugging these values into the formula, we have:
t = -96 / (2*(-16))
t = -96 / (-32)
t = 3 seconds
b. To find the maximum height, substitute this value of t into the equation s(t):
s(3) = -16(3)^2 + 96(3) + 200
s(3) = -16(9) + 288 + 200
s(3) = -144 + 288 + 200
s(3) = 344 feet
Therefore, the maximum height of the ball is 344 feet.
c. To determine how long it takes before the ball reaches the ground, we need to find the time when the height (s(t)) is equal to zero.
Setting s(t) = 0 in the original equation:
-16t^2 + 96t + 200 = 0
We can solve this quadratic equation for t by factoring or using the quadratic formula. After solving, we find that the ball will reach the ground after approximately 6.5 seconds.
Hence, it will take around 6.5 seconds before the ball reaches the ground.