Question 3
Let $x,y,z$ be nonnegative numbers satisfying
$$x+by\le36,\qquad 2x+3z\le72,$$
where $b$ is a given positive number. Prove that the largest possible value of $x+y+z$ is $\max\{36,\ 24+36/b\}$.
Application. Goods are to be transported from $A$ to $M$ in three ways:
(a) by motor vehicle along $ACM$;
(b) by boat along $ADM$;
(c) by motor vehicle from $A$ to $B$, then by boat from $B$ to $M$.
There are 9 motor vehicles and 45 boats available. Each vehicle on $AB$ carries an average of four tons per day from $A$ to $B$; on $ACM$ it carries 2 tons per day from $A$ to $M$. Each boat on $BM$ carries $8/15$ ton per day from $A$ to $M$ [the route and endpoints are as printed]. How should transport be organized (tons per day by each of the three ways) to maximize total tonnage? How should it be organized if each vehicle on $ACM$ guarantees less than $4/3$ per day [the unit is omitted in this final phrase]?
Editorial note: The source gives no separate boat capacity on $ADM$ and its boat sentence has inconsistent endpoints. Its final phrase says less than $4/3$, not unambiguously equal to $4/3$; the solution distinguishes these cases. Keep this issue in mind when interpreting the application; no missing capacity is silently supplied.
$$x+by\le36,\qquad 2x+3z\le72,$$
where $b$ is a given positive number. Prove that the largest possible value of $x+y+z$ is $\max\{36,\ 24+36/b\}$.
Application. Goods are to be transported from $A$ to $M$ in three ways:
(a) by motor vehicle along $ACM$;
(b) by boat along $ADM$;
(c) by motor vehicle from $A$ to $B$, then by boat from $B$ to $M$.
There are 9 motor vehicles and 45 boats available. Each vehicle on $AB$ carries an average of four tons per day from $A$ to $B$; on $ACM$ it carries 2 tons per day from $A$ to $M$. Each boat on $BM$ carries $8/15$ ton per day from $A$ to $M$ [the route and endpoints are as printed]. How should transport be organized (tons per day by each of the three ways) to maximize total tonnage? How should it be organized if each vehicle on $ACM$ guarantees less than $4/3$ per day [the unit is omitted in this final phrase]?
Editorial note: The source gives no separate boat capacity on $ADM$ and its boat sentence has inconsistent endpoints. Its final phrase says less than $4/3$, not unambiguously equal to $4/3$; the solution distinguishes these cases. Keep this issue in mind when interpreting the application; no missing capacity is silently supplied.