2023 AMC 10 B Problem 24

Image of a Cube and Convex Hull Method

Since the coordinate equations are linear and the variables are bounded between $0$ and $1$, the resulting region is a flat convex polygon. The vertices of this boundary must come from testing the bounds where the parameters $u,v,$ and $w$ are exactly $0$ or $1$.

Step 1: Understand the region as a transformed geometric solid

The provided coordinates describe a linear transformation applied to a point $(u,v,w)$ chosen from the 3D unit cube. Because linear transformations map convex objects to convex objects, the boundary of the resulting 2D region is a convex polygon. The vertices of this polygon must be the images of the 8 corner vertices of the original 3D cube.

Step 2: Calculate the coordinates of the 8 transformed vertices

Substitute the 8 possible extreme combinations of the independent variables into the expression $(2u-3w,v+4w)$ to find the corresponding points on the coordinate plane.

Step 3: Identify the extreme horizontal and vertical edges

We identify four outer edges by finding the minimum and maximum coordinate values among the 8 generated points. The minimum $y$-coordinate is $0$, connecting $(0,0)$ and $(2,0)$. The maximum $y$-coordinate is $5$, connecting $(-3,5)$ and $(-1,5)$. The maximum $x$-coordinate is $2$, connecting $(2,0)$ and $(2,1)$. The minimum $x$-coordinate is $-3$, connecting $(-3,4)$ and $(-3,5)$. We calculate the lengths of these four outer edges.

Step 4: Verify the connecting diagonals to complete the boundary

The four extreme edges leave two gaps in the perimeter: one from $(2,1)$ to $(-1,5)$, and another from $(-3,4)$ to $(0,0)$. We must verify that these two connecting segments truly enclose all other points. The line connecting $(0,0)$ and $(-3,4)$ has the equation $4x+3y=0$. Plugging in the other 6 points yields values greater than or equal to $0$, confirming they all lie on one side of this line. The line connecting $(2,1)$ and $(-1,5)$ has the equation $4x+3y=11$. Plugging in the other 6 points yields values less than or equal to $11$, confirming it is the final boundary. We calculate the lengths of these two diagonals.

Step 5: Calculate the total perimeter

Add the lengths of the six verified boundary edges to find the total perimeter of the region.

Vector Addition and Minkowski Sum Method

The coordinates can be separated into three independent vectors being added together, each scaled by a fraction between $0$ and $1$. This geometric operation builds a shape whose perimeter depends directly and simply on the lengths of those three generating vectors.

Step 1: Express the coordinates as a sum of scaled vectors

Notice that the expressions for the $x$ and $y$ coordinates can be separated by the independent variables $u,v$, and $w$. We can rewrite any point in the region as a vector addition.

Step 2: Recognize the geometric shape as a Minkowski sum

Because $u,v,$ and $w$ each independently range from $0$ to $1$, the region is formed by adding together all possible points on three independent line segments. This operation is known as a Minkowski sum, and it generates a central-symmetric convex polygon called a zonogon.

Step 3: Calculate the lengths of the generating vectors

Find the magnitude of each of the three generating vectors using the Pythagorean theorem.

Step 4: Apply the zonogon perimeter property

Because each vector appears twice on the boundary of the final polygon, the total perimeter is simply twice the sum of the lengths of the three generating vectors.