Answer:
11.5 cm²/s
Step-by-step explanation:
We must solve two problems to answer this question:
For the first problem, we have ...
... 9 = 5t +4√t
... 9 -5t = 4√t . . . . . subtract 5t
... 81 -90t +25t² = 16t . . . . square both sides
... 25t² -106t +81 = 0 . . . subtract 16t
... (t -1)(25t -81) = 0 . . . . factor
... t = 1 or 3.24 . . . . . . t=3.24 is an extraneous solution
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For the second problem, we have area (a(t)) is ...
... a(t) = length×height = (5t +4√t)(√t) = 5t^(3/2) +4t
Then the derivative is ...
... a'(t) = (3/2)(5√t) +4
and
... a'(1) = (3/2)(5√1) +4 = 11.5 . . . . . cm²/s
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Comment on the attachment
A graphing calculator can help solve both problems. It can help find the time at which length is 9 cm by solving ... length-9 = 0. (The nice thing here is that there is no extraneous solution.) It can compute the derivative of a function, too.