Question 595·Hard·Algebra

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A warehouse prepares shipments using small and large crates. Each small crate holds $4$ cubic units and requires $2$ minutes to prepare. Each large crate holds $7$ cubic units and requires $5$ minutes to prepare.

If $x$ is the number of small crates and $y$ is the number of large crates, the shipment must hold at least $120$ cubic units and require no more than $80$ minutes to prepare. Both $x$ and $y$ are nonnegative integers.

Which choice gives the greatest possible value of $y$ ?

A system of inequalities models a shipment with both a minimum capacity and a maximum preparation time. Graph the inequalities in Desmos and look at the highest overlap to find which large-crate counts are worth testing. The graph includes fractional coordinates, though, so its top point may not describe a real shipment. Check whole-number crate counts near that point against both limits.

Hints

Hint 1
For capacity, multiply each crate count by the space that kind holds, then add. For preparation time, do the same with minutes. Which inequality symbol matches at least, and which matches no more than?
Hint 2
The overlap is where both shaded regions meet, so every point there passes both limits. Look at its highest point to find an upper bound on $y$ , but remember that crate counts must be whole numbers.
Hint 3
Test the highest plausible whole-number $y$ first. A horizontal line through that height shows which $x$ values satisfy both limits. Does that interval contain a whole-number $x$ ?

Step-by-step

Graph the limits, then check whole-number points
Step 1
Translate the shipment limits
The small crates contribute $4x$ cubic units and the large crates contribute $7y$ , so at least $120$ gives $4x+7y\ge120$ . Preparation takes $2x+5y$ minutes, so no more than $80$ gives $2x+5y\le80$ . Also, $x$ and $y$ must be nonnegative integers.
Step 2
Find the highest part of the overlap
Type the four inequalities into Desmos. The overlap is where every shaded region meets. Click where the capacity and time boundary lines cross: Desmos shows about $(6.667,13.333)$ . This is the highest point of the overlap, so $y\ge14$ cannot work. The largest whole-number height still worth testing is $13$ .
Step 3
Rule out the highest candidate
Add $y=13$ and click where it meets each boundary of the overlap. Desmos shows $(7.25,13)$ and $(7.5,13)$ , so $x$ would have to be between $7.25$ and $7.5$ . There is no whole-number $x$ in that interval. Only integer points in the overlap represent shipments.
Step 4
Check the next whole-number count
Try $y=12$ . Those large crates hold $7(12)=84$ cubic units, leaving $120-84=36$ ; $9$ small crates supply $4(9)=36$ . Type $(9,12)$ and both totals. Desmos plots the point in the overlap and prints $120$ cubic units and $78$ minutes. Both limits hold, and every greater whole-number $y$ has been ruled out. The greatest possible number of large crates is $12$ . Choice C.

Question 595·Hard·Algebra

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