Year: 1994
Paper: 2
Question Number: 8
Course: LFM Stats And Pure
Section: Curve Sketching
No solution available for this problem.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1516.0
Banger Comparisons: 1
`24 Hour Spares' stocks a small, widely used and cheap component.
Every $T$ hours $X$ units arrive by lorry from the wholesaler, for
which the owner pays a total $\pounds (a+qX)$. It costs the owner $\pounds b$
per hour to store one unit. If she has the units in stock she expects
to sell $r$ units per hour at $\pounds(p+q)$ per unit. The other running
costs of her business remain at $\pounds c$ pounds an hour irrespective
of whether she has stock or not. (All of the quantities $T,X,a,b,r,q,p$
and $c$ are greater than 0.) Explain why she should take $X\leqslant rT$.
Given that the process may be assumed continuous (the items are very
small and she sells many each hour), sketch $S(t)$ the amount of
stock remaining as a function of $t$ the time from the last delivery.
Compute the total profit over each period of $T$ hours. Show that,
if $T$ is fixed with $T\geqslant p/b$, the business can be made
profitable if
\[
p^{2}>2\frac{(a+cT)b}{r}.
\]