into the air, and its height h in feet after t seconds is given by h left parenthesis t right parenthesis equals negative 16 t squared plus 96 t plus 3h(t)=−16t2+96t+3.
(a) What is the height of the baseball when it is hit?
(b) After how many seconds does the baseball reach its maximum height?
(c) Determine the maximum height of the baseball.
herding my way through that first "sentence" ....
I got the obvious:
h = -16t^2 + 96t + 3
a) plug in t = 0
b) find the vertex and both b) and c) are answered.
To find the answers to these questions, we need to analyze the given quadratic equation and understand its properties.
The equation h(t) = -16t^2 + 96t + 3 represents the height of the baseball as a function of time. Let's break down each part of the equation:
- The coefficient of t^2 (-16) indicates that the baseball is affected by gravity, causing it to follow a parabolic trajectory.
- The coefficient of t (96) represents the initial upward velocity, in feet per second, of the baseball when it is hit.
- The constant term (3) represents the initial height of the baseball when it is hit.
Now let's answer each part of the question step by step:
(a) What is the height of the baseball when it is hit?
To determine the height of the baseball when it is hit, we can substitute t = 0 into the equation h(t). So, plugging in t = 0, we get:
h(0) = -16(0)^2 + 96(0) + 3
h(0) = 0 + 0 + 3
Therefore, the height of the baseball when it is hit is 3 feet.
(b) After how many seconds does the baseball reach its maximum height?
To find the time when the baseball reaches its maximum height, we need to find the x-coordinate of the vertex of the parabolic graph described by the equation h(t).
The x-coordinate of the vertex can be found using the formula: t = -b / (2a), where a and b are the coefficients of t^2 and t respectively in the equation h(t).
In our equation h(t) = -16t^2 + 96t + 3, a = -16 and b = 96. Plugging these values into the formula, we get:
t = -96 / (2 * -16)
t = -96 / -32
t = 3
Therefore, the baseball reaches its maximum height after 3 seconds.
(c) Determine the maximum height of the baseball.
To find the maximum height of the baseball, we substitute the value of t we found in part (b) back into the equation h(t).
Plugging t = 3 into h(t) = -16t^2 + 96t + 3, we get:
h(3) = -16(3)^2 + 96(3) + 3
h(3) = -16(9) + 288 + 3
h(3) = -144 + 288 + 3
Therefore, the maximum height of the baseball is 147 feet.