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.
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!!!!!
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.