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1.Suppose that the total number of units produced by a worker in t hours of an 8 hour day can be modeled by
P(t) = 25t + 6t2 ? t3a) Find the point of diminishing returns. Explain what this point means in terms of units produced by a worker.
2.The weekly sales S of a product during an advertising campaign are given by
, 0 ? t ? 20
a)Use calculus to find the critical values that lie in the domain of the problem.b)Over what interval in the domain do the sales decrease?c)Over what interval in the domain do the sales increase?
d)When will the sales be a maximum?
e)What will the maximum sales be?
f)Graph the function and label the critical values on the graph. (You may want to this after part a.)
1.P(t) = 25t + 6?t ?^2 ? t^3
dP/dt=25+12t-3?t ?^2 25+12t-3?t ?^2 yields
t=5.51 and-1.51
But we can’t have negative time. Next we check whether it is a minimum or a maximum
(d^2 P)/(dt^2 )=12-6t<0,t=5.51 is maximum
a) Thus Diminished return
t=5.51
Maximum units = 25(5.51)+ 6(5.51)^2 ? (5.51)^3=152.63≈153
This point means that for maximum units to be produced a worker has to work for 5.51 hours and the maximum units at this time is 153 units.
4 a)To find the critical values then we have to find the derivative of S(t)
S^' (t)=((t^2+25) d/dt (26t)-26t d/dt (t^2+25))/(t^2+25)^2
=((t^2+25)26-26t(2t))/(t^2+25)^2
=(650-26t^2)/(t^2+25)^2
To get critical valuesS^' (t)=0. Thus
(650-26t^2)/(t^2+25)^2 =0
650=26t^2
t^2=650/26=25
t=5 read more.
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