# The marginal productive output of workers in a small manufacturing firm is given by MP = 8L-L^2+20 where L is the number of workers hired by the firm. Rewrite the equation above as a first order differential equation and find the general solution for the total productive output function P(L).

The marginal productive output of workers in a small manufacturing firm is given by $MP=8L-{L}^{2}+20$ where L is the number of workers hired by the firm. Rewrite the equation above as a first order differential equation and find the general solution for the total productive output function P(L).
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Willie

Given that marginal productive output of workers in a small manufacturing firm is given by $MP=8L-{L}^{2}+20$
Since marginal productive output of workers in a small manufacturing firm is given by $MP=8L-{L}^{2}+20$ that is $dP/dL=8L-{L}^{2}+20$
Find the general solution for the total productive output function P(L)
$P=\int \left(8L-{L}^{2}+20\right)dL$
$=\left[\left(8{L}^{2}\right)/2-{L}^{3}/3+20L\right]$
$=4{L}^{2}-{L}^{3}/3+20L+c$
Thus, the general solution for the total productive output function
$P=4{L}^{2}-{L}^{3}/3+20L$