Designing Under Constraints: Practice

A grounds crew compares three rectangular areas with a $180$ dollar edging budget. Edging costs $6$ dollars per meter on all four sides, and all layouts fit the site. The proposals are $6$ m by $8$ m, $5$ m by $10$ m, and $7$ m by $9$ m. Which gives the most area within budget?

A supplier has two box designs. Each meets the same capacity requirement. Design P uses $140$ square centimeters of board and has a $15$ cent fee. Design Q uses $155$ square centimeters of board and has a $5$ cent fee. Board costs $0.4$ cents per square centimeter and there is no waste. Which is cheaper and by how much?

A tray is $31$ cm wide and $22$ cm deep. Each card is $7$ cm by $4$ cm. Pack the cards in a rectangular grid in one layer with all cards oriented the same way and edges parallel to the tray. They can touch. Which orientation holds more?

A raised bed requires $15$ m of fence on two sides at $7$ dollars per meter and $11$ m total on the other two at $5$ dollars per meter. One of the cheaper sides is $5.5$ m long and already has usable fence. What will the new fence cost?

A supplier estimates total charge for a box at $96$ cents before waste. A competing box costs $103$ cents and requires no waste allowance. Extra material for the first box costs $0.35$ cents per square centimeter. How much extra area would make the costs tie?

A parcel service offers the four sizes of closed rectangular box shown below. All sizes cost the same per square centimeter of board, with no waste or fees. For an order needing at least $90\text{ cm}^3$, which size minimizes cost?

An outdoor seating plan must provide at least $36$ seats and leave a clear aisle $2$ m or wider. Among feasible plans, the lowest rental cost wins. Plan K has $36$ seats, a $2$ m aisle, and a $210$ dollar rental. Plan L has $42$ seats, a $1.8$ m aisle, and a $175$ dollar rental. Plan M has $40$ seats, a $2.3$ m aisle, and a $215$ dollar rental. Which plan should be chosen?

A room has the same square tiles in every complete row. Each row fits up to $7$ tiles, and no more than $9$ rows fit. A layout uses $8$ rows of $7$ tiles, and the whole set of $9$ full rows is permitted. What can be certified for this row layout family?

A rehearsal platform is $10$ m by $8$ m. A clear strip runs along the full $10$ m side and leaves the rest of the floor for performers. How much usable area is lost if the strip must widen from $1$ m to $1.6$ m?

A box must be $3$ cm high, hold exactly $60$ cubic centimeters, and be at most $4$ cm wide. Length and width can be any positive real numbers that meet those requirements. Which statement correctly describes the feasible base dimensions?

A sample tests $2$ m by $13$ m, $4$ m by $9$ m, and $6$ m by $5$ m panels. Each tested panel meets the only constraints, whole-meter sides and a perimeter of at most $30$ m. Which permitted panel proves that the best tested area of $36$ square meters is not optimal?

Two possible produce trays cost $48$ and $60$ cents per tray and hold $12$ and $16$ pieces respectively. Both fit the storage rack. The objective changes from minimizing cost per tray to minimizing cost per piece at full capacity. Which tray now wins?