Question

Pomegranate is thrown from ground level straight up into the air at time t=0 with velocity 176 feet per second. Its height in feet at t seconds is f(t)=−16t2+176t. Find the time it hits the ground and the time it reaches its highest point.

Answer 1

The pomegranate hits the ground at t = 0 seconds and t = 11 seconds. The pomegranate reaches its highest point at t = 5.5 seconds.

Step-by-step explanation:

To find the time the pomegranate hits the ground, we need to find the value of t when the height function f(t) equals 0. Setting -16t² + 176t = 0, we can factor out -16t to get: t(-16t + 176) = 0. This equation is true when either t = 0 or -16t + 176 = 0. Solving -16t + 176 = 0, we find t = 11. The pomegranate hits the ground at t = 0 seconds and t = 11 seconds.

To find the time the pomegranate reaches its highest point, we can find the vertex of the parabolic function f(t) = -16t² + 176t.

The formula for the x-coordinate of the vertex is given by t = -b/(2a), where a = -16 and b = 176. Plugging in these values, we find t = -176/(2*(-16)) = 5.5. Therefore, the pomegranate reaches its highest point at t = 5.5 seconds.

Answer 2

Position is given as function of time

`h = -16t^2 + 176 t`

now rate of change in position with time is given as

`v = (dh)/(dt)`

`v = -32 t + 176`

so when it reached to highest point its velocity will be zero

`0 = -32*t + 176`

`t = 5.5 s`

so it will take 5.5 s to reach the highest point

Also we know that it took same time to reach back on ground too

So total time taken by the object to come back is given by

`T = 5.5 + 5.5 = 11 s`

so it will take 11 s to reach back on ground