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Designing Under Constraints: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

A panel shop can use only rectangular dimensions, in whole centimeters, with both sides at least $3$ cm. A board edge constraint requires length plus width to be at most $13$ cm. With length named as the longer side, which panel has the proven greatest area?

Answer choices for question 1 A $8$ cm by $5$ cm B $7$ cm by $7$ cm C $9$ cm by $4$ cm D $7$ cm by $6$ cm
2

One shipping shelf has room for $24$ cm across and $18$ cm front to back. Pack $7$ cm by $4$ cm cartons with aligned edges in horizontal rows. Each row uses a single orientation, but rows may differ. Cartons can touch and fit in one layer. What is the proven maximum for this family?

Answer choices for question 2 A $12$ B $16$ C $15$ D $18$
3

A carton has a $9$ cm by $6$ cm base and height $4$ cm. It needs a lid and a full-height partition spanning the $6$ cm side. Ignore thickness and waste. What area of board is needed for the six outside faces and the partition?

Answer choices for question 3 A $228\text{ cm}^2$ B $252\text{ cm}^2$ C $276\text{ cm}^2$ D $240\text{ cm}^2$
4

A museum display case has fixed inside height and base area. In its model, the board to be minimized has area $C+2u+5v$ , where $C$ is fixed, $u$ and $v$ are positive base lengths in centimeters, and $uv=40$ . All positive real lengths satisfying this product are allowed. Which dimensions attain the minimum, with the length $u$ given first?

Answer choices for question 4 A 10 cm by 4 cm B 4 cm by 10 cm C 8 cm by 5 cm D 20 cm by 2 cm
5

Two tent plans require $28$ and $31$ square meters of fabric respectively, before cutting waste. Waste adds $15\%$ of the first plan's required area and $4\%$ of the second plan's. With at most $32.3$ square meters to purchase, which plans are feasible?

Answer choices for question 5 A Only the first plan B Only the second plan C Both plans D Neither plan
6

An arts fair has room for exactly $3$ equal vendor rows. Row lengths can be $11$ m or $14$ m. All booths are $2$ m wide along a row, with a $1$ m gap between neighbors and no required gap at the ends. How many more booths fit in total if the rows are lengthened from $11$ m to $14$ m?

Answer choices for question 6 A $3$ B $1$ C $6$ D $0$
7

A metal rack has four allowed rectangular base sizes. A $6$ cm by $8$ cm, B $4$ cm by $11$ cm, C $5$ cm by $10$ cm, and D $7$ cm by $7$ cm. It needs a base area of at least $47$ square centimeters. A new shelf allows a base to be turned but requires one side to be at most $6$ cm and the other at most $9$ cm. Among feasible sizes, which has least perimeter?

Answer choices for question 7 A B B A C C D D
8

A contractor can use either of two screening designs with the same enclosed area. Plan R needs $45$ m of screen and has a $70$ dollar fitting fee. Plan S needs $55$ m and has a $30$ dollar fee. What is the complete range of screen prices per meter for which R is cheaper?

Answer choices for question 8 A Any price below $4$ dollars per meter B Exactly $4$ dollars per meter only C Any price above $10$ dollars per meter D Any price above $4$ dollars per meter
9

A packing table shows that $19$ cards is the best count in full-width rows. An independent area calculation shows that at most $19$ cards of this size can fit inside the container without overlap, even in other orientations. What follows?

Answer choices for question 9 A $19$ is the greatest possible count among all permitted nonoverlapping arrangements. B $19$ is best only within full-width rows, even with the area bound. C $20$ might fit in another orientation. D No optimum is established until all arrangements are individually listed.
10

A sign uses one sheet with at most $165$ square centimeters available. A closed rectangular case for it needs a base and lid of $35$ square centimeters each, four side faces totaling $80$ square centimeters, and an allowance of $10\%$ of the face area for waste. Which statement describes the area requirement?

Answer choices for question 10 A The calculated area leaves $15$ square centimeters of the allowance unused. B The calculated area equals the full sheet allowance. C The design exceeds the sheet limit by $10$ square centimeters. D The design exceeds the sheet limit by $15$ square centimeters.
11

A bin is $25$ cm deep. It uses one layer of nonoverlapping rows, which may be $6$ cm or $9$ cm deep. A $6$ cm row holds $4$ pots at most, and a $9$ cm row holds $7$ at most. All full rows that meet the depth limit are permitted. If the available depth drops to $24$ cm, what happens to the maximum?

Answer choices for question 11 A It drops from $18$ to $16$ pots. B It drops from $21$ to $18$ pots. C It rises from $16$ to $18$ pots. D It stays at $18$ pots.
12

For two positive panel dimensions $x$ and $y$ , the material charge in a model is $5(x+y)$ dollars, with $x$ and $y$ in centimeters. The design requires $xy=81$ . A test finds a feasible pair with charge $90$ dollars. What argument proves this charge is a minimum?

Answer choices for question 12 A A fixed product makes the sum the same for every pair. B Some tested pairs have a larger sum, so every other pair does too. C The square $(x-y)^2$ is nonnegative, so $(x+y)^2$ is at least $324$ and the positive sum is at least $18$ . D The two dimensions both have to be positive, so each must be at least $9$ cm.

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