A pomegranate is thrown from ground level straight up into the air at time t=0 with velocity 128 feet per second. Its height in feet at t seconds is f(t)=16t^2+128t. Find the time its hits the ground and the time it reaches its highest point.
To find the time the pomegranate hits the ground, we need to find the value of t when the height is 0.
Given the function f(t) = 16t^2 + 128t, we can set it equal to 0 and solve for t:
16t^2 + 128t = 0
Factoring out a common factor of 16t:
16t(t + 8) = 0
Setting each factor equal to 0 and solving for t:
16t = 0, t + 8 = 0
From the first equation, we can see that t = 0.
From the second equation, we can see that t = -8.
Since time cannot be negative in this context, we can discard the solution t = -8.
Therefore, the pomegranate hits the ground at t = 0 seconds.
To find the time the pomegranate reaches its highest point, we need to find the vertex of the quadratic function f(t) = 16t^2 + 128t.
The vertex of a quadratic function in the form f(t) = at^2 + bt + c is given by t = -b / (2a).
In this case, a = 16 and b = 128.
t = -128 / (2 * 16) = -128 / 32 = -4
Therefore, the pomegranate reaches its highest point at t = -4 seconds.
Note that the time cannot be negative in this physical context, so we discard the solution t = -4.
Therefore, the pomegranate reaches its highest point at t = 4 seconds.
To find the time when the pomegranate hits the ground, we need to determine when its height, f(t), is equal to zero. In other words, we need to solve the equation:
16t^2 + 128t = 0
To factor out t from this equation, we can divide both sides by 16:
t^2 + 8t = 0
Now, we can factor out a common term of t:
t(t + 8) = 0
Setting each factor equal to zero, we have two possibilities:
t = 0 or t + 8 = 0
Solving t + 8 = 0, we find t = -8. Since time cannot be negative in this case, we discard this solution.
Therefore, the pomegranate hits the ground when t = 0.
Your equation should be
f(t) = - 16t^2 + 128t
so
-16t^2 + 128t = 0
t^2 - 8t = 0
t(t-8) = 0
t = 0 ----> when you tossed it
t = 8 ----> when it lands
so it reached the hightest point at the midpoint of that 8 seconds, namely at 4 seconds
f(4) = -16(4^2) + 128(4)
= 256 ft
Wow!, That is quite a feat to through a pomegranate 256 feet into the air!!!!!