Solving a function for a variable, confusion

Question

I have the function $f(t) = -4.9t^2+25t+3$, where $f(t)$ is a the height of a grapefruit after $t$ number of seconds.

I need to find out how long the grapefruit is in the air, so I know i need to solve for t by setting $f(t)$ to zero.

But I'm confused about how to go about doing this

Answer 1

When it reaches the max height its at the highest point. This means $f'(t) = 0 \to -9.8t + 25 = 0\to t = \frac{25}{9.8} = 2.55$ sec.

Answer 2

You need to solve $f(t) = 0$. That is:

$$f(t) = -4.9t^2+25t+3 = 0 \Rightarrow 4.9t^2-25t-3 = 0 $$ Solutions are: $$t_{1} = \frac{25-\sqrt{25^2-4\cdot 4.9 \cdot (-3)}}{2\cdot 4.9}= -0.1173$$ $$t_{2} = \frac{25+\sqrt{25^2-4\cdot 4.9 \cdot (-3)}}{2\cdot 4.9}= 5.2193$$

Assuming that the launch start at $t=0$, then the first solution makes no sense since $t_1 < 0$.

Then $t_2$ is the time when the height $=0$, that is, the grapefruit hits the ground. So, your solution is $5.2193 s$.

Answer 3

The problem is not so clear. At $t=0$ the height is $3$m so, it is in the air or not?. Depending on the answer you have:

If NO, then solve the equation $f(t)=0$ for t and find the positive solution. This is the request time.

If YES, then solve the same equation but the answer now is the interval between the two solutions. I.e. :the grapefruit has begun his fly before $t=0$.