Answer: Maximum height = 105 feet
And, It takes 9/4 seconds to reach that point.
Step-by-step explanation:
Here the given function that shows the height of the pumpkin,
[tex]h(t) = -16t^2 + 72t + 24[/tex] --------(1)
Where t is the time in second.
Differentiating equation (1) with respect to t,
We get, [tex]h'(t) = -32 t + 72[/tex]
Again differentiating above equation with respect to t,
We get, [tex]h''(t) = -32 [/tex]
For maximum or minimum, [tex]h'(t) = 0[/tex]
[tex]- 32 t + 72 = 0[/tex]
[tex]32 t = 72[/tex]
[tex]t = \frac{9}{4}[/tex]
At t = 9/4 , h''(t) = Negative value,
Therefore, At t = 9/4 seconds, h(t) is maximum,
And, the maximum value is,
[tex]h(\frac{9}{4} ) = -16(\frac{9}{4})^2 + 72(\frac{9}{4}) + 24[/tex]
[tex]h(\frac{9}{4} ) = -16(\frac{81}{16})+ 72(\frac{9}{4}) + 24[/tex]
[tex]h(\frac{9}{4} ) = -81 + 162 + 24=105[/tex]
Therefore, the maximum height of the pumpkin is 105 feet at 9/4 seconds
The maximum height 105 feet and the time it takes to reach this point is 2.25 seconds.
Given that the pumpkin height (h) at time t is given by:
h(t) = -16t²+ 72t +24
The maximum height of the pumpkin is at dh/dt = 0, hence:
dh/dt = -32t + 72
-32t + 72 = 0
32t = 72
t = 2.25 seconds
Therefore the maximum height is at 2.25 seconds. Hence:
h(2.25) = -16(2.25)² + 72(2.25) +24 = 105 feet
The maximum height 105 feet and the time it takes to reach this point is 2.25 seconds.
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