Question

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(1) A mixing tank is filled and drained in gallons) after t hours according to f(t) = 9ť – (a) For which time intervals is the volume increasing or decreasing? When does the maximum occur and how many gallons is it? (5 points) (b) For which time intervals is the filling or draining rate increasing or decreasing (i.e. concavity)? At which time is the tanking being filled the fastest (i.e. inflection point)? (5 points)

Answer 1

Given,

Volume of tank after time t as:

$f(t) =9t^2 - \frac{3}{2}t^4$

For volume f(t) to be increasing or decreasing, rate of change of f(t) w.r.t. time must be positive or negative respectively.

And, for the rate of filling and draining to be increasing and decreasing, the second derivative of f(t) w.r.t. time must be positive and negative respectively.

Now,

0 Giveni flt) = 2 =9t² t4 gallons in the tank after t howres. Here, flt) is the @ For the volume to increase :- dflt) => (97-344) >0 18t-l2t> > at dt

=> 18+ 3 -6 t > 0 > 6+(3-42) >O G+(3++) (3-7) >O > Gt 1/3+t) Ct-3)<o and +20 l as time is always positiva) + + म 3 V3 Thus, t E B, V5) ferr flt) to be increasing

- For volume to decrease : dflt) so : => d (94-344) <0 at 6+T 13++)613++) >O (see part @ + + -S 33 0 E These for volume to be decreasing, time, t t (B, 8) SO and -0 Tihen masumum occurs, df(t) DFD at LO -> dfttt df 9 + ²-3/40 => T87-6ť at >6013- +2) =O → t=0, +13,- 13 and dt?