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 $0 \le w \le 1?$

$10\sqrt{3}$

$10$

$12$

$18$

$16$

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 $(-3w, 4w).$ Now let $w$ run from $0$ to $1.$ The rectangle slides along the vector $(-3, 4),$ which has length $5.$ So the region is the Minkowski sum of that rectangle and the segment, and its perimeter is the rectangle's perimeter plus twice the segment length: $2(2 + 1) + 2 \cdot 5 = 16.$ Therefore, the answer is E.

2023 AMC 10B Problem 24

Solution:

Problem 24 in Other Years