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Question

# For a business using just-in-time inventory, a delivery of Q items arrives just as the last item is shipped out. Suppose that items are shipped out at a nonconstant rate such that $f(t)=Q-r \sqrt{t}$ gives the number of items in inventory. Find the time T at which the next shipment must arrive. Find the average value of f on the interval [0, T].

Solution

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The inventory will need to be restocked when

$f(T) = 0 \implies Q-r\sqrt{T} = 0 \implies T = \frac{Q^2}{r^2} \tag{1}$

The average value of $f$ in $[0,T]$ is given by

$f_{avg} = \frac{1}{T-0} \int_0^T f(t) \; \text{d} t$

The integral is

$\int_0^T f(t) \; \text{d} t = \int_0^T Q-r\sqrt{t} \; \text{d} t = \left[ Qt - \frac{2}{3} r t^{3/2} \right|_{t=0}^{t=T} = QT - \frac{2}{3} r T^{3/2}$

so

$f_{avg} = Q - \frac{2}{3} r \sqrt{T} \overset{(1)} = Q - \frac{2}{3} \; r \; \frac{Q}{r} = \frac{Q}{3}$

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