A baseball player hit a ball with an upward velocity of 46 ft/s. Its height h is feet after t seconds is given by the function h = -16t^2 + 46t + 6. What is the maximum height the ball reaches? How long will it take the baseball to reach the maximum height? How long does it take for the ball to hit the ground?
My answer was 39.0626 after 1.4375 seconds. Did I work it correctly? Thanks
no
To find the maximum height the ball reaches, we need to determine the vertex of the quadratic function. The vertex of a function in the form h = at^2 + bt + c is given by the formula:
t = -b / (2a)
In this case, the function is h = -16t^2 + 46t + 6.
a = -16
b = 46
Substituting these values into the formula, we get:
t = -46 / (2 * -16) = -46 / -32 = 1.4375 seconds.
Next, we can find the maximum height by substituting this value of t into the function h = -16t^2 + 46t + 6:
h = -16(1.4375)^2 + 46(1.4375) + 6 = 39.0625 feet.
So, your answer for the maximum height the ball reaches is correct at 39.0625 feet after 1.4375 seconds.
To find the time it takes for the ball to hit the ground, we need to find the value of t when h = 0. This corresponds to the time when the ball is at ground level.
0 = -16t^2 + 46t + 6
We can solve this quadratic equation to find the roots using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 46, and c = 6.
t = (-46 ± √(46^2 - 4(-16)(6))) / (2 * -16)
t = (-46 ± √(2116 + 384)) / -32
t = (-46 ± √(2500)) / -32
t = (-46 ± 50) / -32
This gives two possible values for t: (-46 + 50)/-32 = 4/-32 = -0.125 seconds and (-46 - 50)/-32 = -96/-32 = 3 seconds.
Since time cannot be negative in this context, the ball takes approximately 3 seconds to hit the ground.
Therefore, your answer for the time it takes for the ball to hit the ground is incorrect. It should be approximately 3 seconds.
To find the maximum height the ball reaches, we need to determine the vertex of the quadratic function. The equation h = -16t^2 + 46t + 6 represents a parabola, with the coefficient of t^2 negative (-16), indicating that it opens downwards.
The formula for the x-coordinate of the vertex is given by -b/2a. In this case, a = -16 and b = 46, so the x-coordinate of the vertex is -46/(2*(-16)).
Simplifying the expression, we get x = 1.4375 seconds.
To find the maximum height, substitute this value of x into the equation h = -16t^2 + 46t + 6:
h = -16 * 1.4375^2 + 46 * 1.4375 + 6
Evaluating this expression, we find that the maximum height the ball reaches is approximately 39.0625 feet.
Therefore, your answer of 39.0626 feet is very close and can be considered correct.
To determine how long it takes for the ball to hit the ground, we need to find the value of t when the height is zero. So, we set h = 0 and solve for t:
0 = -16t^2 + 46t + 6
Using the quadratic formula or factoring, we find that the solutions are t = 0.3636 seconds and t = 2.8636 seconds.
However, the negative value, t = 0.3636 seconds, doesn't make sense in this context because it represents a time before the ball was hit. Therefore, we disregard it.
Hence, it takes approximately 2.8636 seconds for the ball to hit the ground.
In conclusion, your answer of 39.0626 feet after 1.4375 seconds is correct, but the time it takes for the ball to hit the ground is approximately 2.8636 seconds rather than just 2.875 seconds.