2011 AMC 10B Problem 3

P1-10~1 minPrint

Absolute Value & Inequalities Quadrilaterals & Polygon Areas Consider the largest/smallest element or boundary case

At a store, when a length is reported as $x$ inches that means the length is at least $x - 0.5$ inches and at most $x + 0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle?

Show solution without answering

Key insight

Each side could be up to half an inch shorter than reported. The smallest area uses the shortest possible width and the shortest possible length together.

Solution

First decide what each reported number really allows.

1. "Reported as $2$ inches" means the true width is somewhere from $1.5$ to $2.5$ inches.
2. "Reported as $3$ inches" means the true length is somewhere from $2.5$ to $3.5$ inches.
3. The area is width times length. Making either side shorter makes the area smaller, so to get the smallest area, use the shortest value for both sides.
4. The smallest area is
$1.5 \times 2.5 = 3.75.$

The answer is $\boxed{\textbf{(A)}\ 3.75}$ .

Why this works

When you multiply two positive numbers, shrinking either one shrinks the answer. So the smallest product comes from the smallest value of each factor, and the largest product from the largest of each. The only real work is reading "reported as $x$ " as a range from $x - 0.5$ to $x + 0.5$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

A reported $2$ inches means the true width is between $1.5$ and $2.5$ inches, and a reported $3$ inches means the true length is between $2.5$ and $3.5$ inches.

The area is the product of two positive quantities, so it is smallest when both are as small as allowed:
$1.5 \times 2.5 = 3.75.$

The minimum area is $\boxed{\textbf{(A)}\ 3.75}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 20: Special Functions (§20.2, §20.5)
Introduction to Geometry, Chapter 8: Quadrilaterals
Introduction to Algebra, Chapter 15: More Inequalities (§15.1, §15.3, §15.4)
Introduction to Geometry, Chapter 9: Polygons (§9.3)