A pomegranate is thrown from ground level straight up into the air at time

t = 0 with velocity 96 feet per second. Its height at time t seconds is f(t) = −16t2 + 96t.
a) Find the time it hits the ground. t=? s
b) Find the time it reaches its highest point. t=?s
c) What is the maximum height? ____ft

I'm sorry, I do not understand how to even start with this problem. While we are given the starting velocity, how do you know what values to plug in for the rest of the unknowns?

f(t) = −16t^2 + 96t

when it hits the ground, f(t) = 0
-16t^2 + 96t = 0
t^2 - 6t=0
t(t-6) = 0
t = 0 , we knew that
or
t = 6

so it hits the ground after 6 seconds

b) find the vertex of f(t) = -16t^2 + 96t

c) see b)

How do you find the vertex? Do you just put it into a graphing calculator?

Wait nevermind! I got it! Thank you so much!

No problem! Let's break down the problem step by step:

a) To find the time the pomegranate hits the ground, we need to determine when its height, f(t), equals zero. So, we can set the equation f(t) = 0 and solve for t.

Given: f(t) = -16t^2 + 96t

Setting f(t) = 0, we have:
-16t^2 + 96t = 0

Now, we can factor out a common term to simplify the equation:
-16t(t - 6) = 0

To solve this equation, we set each factor equal to zero:
-16t = 0 or (t - 6) = 0

Solving for t, we get:
t = 0 (ignore this since time cannot be negative) or t = 6

Therefore, the pomegranate hits the ground at t = 6 seconds.

b) To find the time the pomegranate reaches its highest point, we need to find the vertex of the parabolic equation f(t) = -16t^2 + 96t. The vertex represents the maximum point of the parabola.

The formula for the x-coordinate of the vertex, in this case, is given by:
t = -b/2a

In our equation, a = -16 and b = 96. Substituting these values:
t = -96 / (2 * -16)
t = -96 / -32
t = 3

Therefore, the pomegranate reaches its highest point at t = 3 seconds.

c) To find the maximum height, we substitute the value of t we found in part (b) back into the equation f(t) = -16t^2 + 96t:

f(3) = -16(3)^2 + 96(3)
f(3) = -16(9) + 96(3)
f(3) = -144 + 288
f(3) = 144

Therefore, the maximum height of the pomegranate is 144 feet.

I hope this helps! Let me know if you have any further questions.