AMC 10 · 2003 · #3
Grade 5 geometry-2d
Pick an answer.
The total cost is a sum of five products (price × area). To make that sum as small as possible you must decide which price lands on which area, so this is a min/max matching problem — Tool #14 (Extreme Principle). The rule it gives is simple: put the cheapest flower on the biggest region and the most expensive flower on the smallest region. Before you can match, Tool #7 (Identify Subproblems) breaks the bed into its five rectangles so each area can be found on its own, and Tool #8 (Analyze the Units) keeps the bookkeeping honest — square feet become number of plants, and plants times dollars-per-plant give dollars.
Find the five region areas
Multiplying each rectangle's sides gives areas 21, 20, 15, 6, 4.
A rectangle's area is just its two side lengths multiplied together.
4.MD.A.3Identify SubproblemsCheapest flower on the biggest region
A bigger region multiplies its price more, so pair the sorted lists in opposite order.
A high price hurts most where it is multiplied by the most plants, so keep high prices on small patches.
A high price hurts most where it is multiplied by the most plants, so dear flowers belong on the small patches.
▸ Why?
A region's bill is one price copied once for every square foot, so the cost is a count of equal groups.
▸ Why?
Swapping a dear flower onto a smaller patch always lowers the bill, and those comparisons chain into one best pairing.
Cost of each region
Multiplying each pair gives costs 21, 30, 30, 15, 12.
Each region's bill is one price copied once for every square foot it covers.
5.NBT.B.7Analyze The UnitsAdd up the least total
Adding them gives 108, and any other pairing costs more, choice (A).
Summing the smartest pairing gives the lowest bill the garden can have.
5.NBT.B.7Extreme PrincipleTo spend the least, put the cheapest flower on the biggest patch and the priciest flower on the smallest patch.
- Find the five region areas
- Cheapest flower on the biggest region
- Cost of each region
- Add up the least total