Porisma

Worked solution

Remove grain before growth and retain the exact supply bound

A grain facility holds 1000010000 bushels at the end of December 2013. Insurance pay-outs purchase extra grain, increasing the stock by 2%2\% before the end of every month.

(a) Find the stock at the end of December 2014. mark

(b) During January–December 2014 the facility supplies x bushels to a laboratory each month, before the 2%2\% addition.

(i) For x=100x=100, show the end-March stock is 10299.910299.9 bushels to 1 d.p. marks

(ii) Find the maximum monthly supply. marks

[2014/HCI/Promo/6]

CJC · Review problems · Question 21

Answer

No-supply year end: 1270012700 bushels (3 s.f.); with supply 100, March stock 10299.910299.9; maximum continuous supply 10000(1.02)12(0.02)1.02[(1.02)12−1]≈927.0548689\dfrac{10000(1.02)^{12}(0.02)}{1.02[(1.02)^{12}-1]}\approx927.0548689, reported as 927927 (3 s.f.) or whole bushels.

(a) Count twelve additions

Without laboratory supply, the year-end stock is 10000(1.02)12≈12682.4179510000(1.02)^{12}\approx12682.41795, or 1270012700 bushels to 3 s.f. as printed.

How did this part go?

(b)(i) Apply withdrawal before multiplication in each month

Put q=1.02q=1.02 and V0=10000V_0=10000. With supply 100100:

MonthBefore growthEnd stock
JanuaryV0−100V_0-100qV0−100qqV_0-100q
FebruaryV1−100V_1-100q2V0−100(q2+q)q^2V_0-100(q^2+q)
MarchV2−100V_2-100q3V0−100(q3+q2+q)q^3V_0-100(q^3+q^2+q)

Thus V3=10000q3−100(q+q2+q3)=10299.9192≈10299.9V_3=10000q^3-100(q+q^2+q^3)=10299.9192\approx10299.9 bushels.

How did this part go?

(b)(ii) Unfold the general monthly stock

For supply x, Vm=q(Vm−1−x)V_m=q(V_{m-1}-x). January, February and March give qV0−qxqV_0-qx, q2V0−x(q2+q)q^2V_0-x(q^2+q) and q3V0−x(q3+q2+q)q^3V_0-x(q^3+q^2+q). In December, V12=10000q12−xq(q12−1)/(q−1)V_{12}=10000q^{12}-xq(q^{12}-1)/(q-1). The source variable-x table has a stray 100100 in one February start entry; the recurrence and general formula use x throughout.

How did this part go?

(b)(ii) Impose feasibility and distinguish continuous from whole-bushel supply

Nonnegative December stock requires

x≤10000q12(q−1)q(q12−1)≈927.0548689.x\le\frac{10000q^{12}(q-1)}{q(q^{12}-1)}\approx927.0548689.

The corresponding bound for month m decreases as m increases, so December is the tightest bound and all earlier withdrawals are feasible.

Monthly supply xDecember stock (approximately)
92614.431
9270.751
928−12.93

The source reports 927 bushels to 3 s.f. This is also the maximum if supply is restricted to whole bushels. If fractional bushels are allowed, the exact expression above is the actual continuous maximum; it must not be replaced by an unqualified exact integer bound.

How did this part go?

Common mistakes

The printed 927 is rounded; continuous and whole-bushel maxima should be distinguished.

Read the hint