Question

What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u - 3w,\ v + 4w)$ with $0 \le u \le 1,$ $0 \le v \le 1,$ and $0 \le w \le 1?$

$10\sqrt{3}$

$13$

$12$

$18$

$16$

Answer: E

Concepts:coordinate geometry vector perimeter

Difficulty rating: 2470

Solution:

Fix $w.$ As $u, v$ sweep $[0, 1]^2,$ the point $(2u - 3w,\ v + 4w)$ fills a $2 \times 1$ axis-aligned rectangle with lower-left corner $(-3w, 4w).$ As $w$ runs from $0$ to $1,$ this rectangle slides along the vector $(-3,4),$ whose length is $5.$ The swept region is a centrally symmetric hexagon. Its opposite pairs of sides have lengths $2, 1,$ and $5,$ inherited from the two sides of the rectangle and the sliding segment. Therefore its perimeter is $2(2+1+5)=16.$ Therefore, the answer is E.

2023 AMC 10B Problem 24

Answer: E

Difficulty rating: 2470

Solution:

Problem 24 in Other Years