Optimizing Water Flow: Finding Maximum Mass and Rate of Change

AI Thread Summary
The discussion focuses on optimizing the mass of water in a leaking container, represented by the function m = 6.00t^0.8 - 3.35t + 23.00. Participants are trying to determine the time at which the water mass is greatest, with the first derivative set to zero to find critical points. There is confusion regarding the correct value of t for maximum mass, with one participant suggesting 2.3375 and others indicating errors in algebra. Additionally, the rate of mass change at specific times (t = 2.00 s and t = 5.00 s) is calculated using the derivative of the mass function. The discussion highlights the importance of careful algebraic manipulation and understanding of derivatives in solving optimization problems.
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Homework Statement



Water is poured into a container that has a leak. The mass m of the water is given as a function of time t by m = 6.00t 0.8 - 3.35t + 23.00, with t 0, m in grams, and t in seconds.
(a) At what time is the water mass greatest?

(b) What is that greatest mass?

(c) In kilograms per minute, what is the rate of mass change at t = 2.00 s?

(d) In kilograms per minute, what is the rate of mass change at t = 5.00 s?

Homework Equations


The first derivative would be 4.8t^-.2-335


The Attempt at a Solution



a)I set the derivative equal to zero and figured t to be 2.3375, but it says that's wrong.
b)I assume I'd plug a into the original and try tht, but I can't get a.
c and d)I tried to put 2 and 5 into t as the origional, but they are not right.
 
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a) is, as you said, the solution for t when

\frac{dm}{dt}=0.

However, the solution to

t^{0.2}=\left[\frac{4.8}{3.35}\right]

is not 2.3375, check your algebra.

b), as you said is m(t) for the solution above
c) and d) can both be found by substituting the times into the expression for \frac{dm}{dt}.
 
Thanks. Where did the 3.35 come from? How would I solve t^-.2?
 
The 3.35 is from the original expression for m, unless you've mistyped it. I used logarithms and the change of base rule to solve for t.
 
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