If I am correct in interpreting the equation as H(t)= 1/12((t^2 -
(t^3)/24)), then I think I have the answer.
To find the maximum value (and minimum value) of any function, we need
to determine their critical values. This can be done with the use of
derivatives.
If we take the first derivative of the function, it becomes
H'(t)= 2t-((t^2)/8)
We then set H'(t)=0 to find the critical points, so
2t-((t^2)/8))=0
t(2-(1/8)t)=0
so t=0 and (2-(1/8)t)=0
...so we find critical points when t=0 and t=16.
Our goal is to find an absolute maximum between one of these two
critical points. If we plug these values in to the second derivative
of the function, we can find the absolute maximum--the value of t
where H''(t) is less than zero. So, the second derivative is
H''(t)= 2-(1/4)t
We plug in our critical values (0 and 16) into H''(t).
H''(0)= 2
H''(16)= -2
Since H''(16) is less than zero, we can conclude that this is an
absolute maximum of the function H'(t), and thus, the tree will reach
its maximum height when t=16. (Additionally, since H''(0) is greater
than 0, we can conclude that t=0 is an absolute minimum.)
Note: You can also take this one step further and find out how tall
the tree will be at that point by plugging 16 in to the original
function.
H(16)=1/12((16^2)-((16^3)/24))
H(16)=64/9
Hope this gets to you in time! |
