h = -4t^2 + 16t + 5?

Question

If a cannon fires so that its height in feet after t = seconds is given by h = -4t^2 + 16t + 5, when is the cannonball at its highest point, and what is the maximum height the cannonball attains?

Answer 1

h(t) = - 4t² + 16 t + 5
h `(t) = - 8t + 16 = 0
t = 2 sec (gives max. height)
h(2) = - 4 (4) + 16(2) + 5
h (2) = - 16 + 32 + 5
h (2) = 21 ft (max. height)

Answer 2

Take the first derivative from -4t^2 + 16t + 5 =
-8t+16
Highest point is at -8t+16=0
-8t=-16
t=-16/-8
t=2 seconds

For maximum height plug 2 in t
h = -4t^2 + 16t + 5
h = 21 feet

Answer 3

To find the maximum height of the ball, you have to find the apex or the crest where the derivative is zero because the tangent at the crest is parallel to the horizontal axis.
So you have to find derivative of h with respect to t and find the value of t where derivative of h vanishes.
h = -4t^2+16t+5
dh/dt = -8t+16
put dh/dt = 0
that means
-8t+16 = 0
or 8t=16
or t = 2

So the ball is at maximum height after 2 seconds and the height after 2 seconds can be had by putting t=2 in the equation h=-4t^2+16t+5 which comes out to be 21 feet.

Answer 4

first find the vertex of the parabola - this will give you the time at which the canonball reaches its max...

Using the vertex formula: t = -b/2a

t = -16/ 2(-4) = -16/-8 = 2

thus the vertex and time it takes for the cball to reach its max height is 2 secs

Now that we have this info, sub it back into the original equation to get the height....

h = -4(2)^2 + 16(2) + 5
h = -4(4) + 32 + 5
h = -16 + 32 + 5
h = 21

Therefore the max height the cball will reach is 21ft.

Hence,

Answer: max height = 21ft at t = 2secs


Calculus way.....

find the second derivative and make that equal to 0 and find t the sub t back into the original equation...

h' = -8t + 16
t = 2

same answer as first method....

Answer 5

t=0 h=5
t=1 h=17
t=2 h=21
t=3 h=17
so, highest point after 2 seconds
maximum height 21

Answer 6

The local minimum is where the dh/dt=0
the dh/dt is -8t+16=0
so it hits a maximum at 2 seconds.