2023 AMC 10B Problems/Problem 24

Solution 1

Notice that this we are given a parametric form of the region, and $w$ is used in both $x$ and $y$ . We first fix $u$ and $v$ to $0$ , and graph $(-3w,4w)$ from $0\le w\le1$ :

$[asy] import graph; Label f; unitsize(0.7cm); xaxis(-5,5,Ticks(f, 5.0, 1.0)); yaxis(-5,5,Ticks(f, 5.0, 1.0)); draw((0,0)--(-3,4)); [/asy]$

Now, when we vary $u$ from $0$ to $2$ , this line is translated to the right $2$ units:

$[asy] import graph; Label f; unitsize(0.7cm); xaxis(-5,5,Ticks(f, 5.0, 1.0)); yaxis(-5,5,Ticks(f, 5.0, 1.0)); draw((0,0)--(-3,4)); draw((2,0)--(-1,4)); [/asy]$

We know that any points in the region between the line (or rather segment) and its translation satisfy $w$ and $u$ , so we shade in the region:

$[asy] import graph; Label f; unitsize(0.7cm); xaxis(-5,5,Ticks(f, 5.0, 1.0)); yaxis(-5,5,Ticks(f, 5.0, 1.0)); draw((0,0)--(-3,4)); draw((2,0)--(-1,4)); filldraw((0,0)--(-3,4)--(-1,4)--(2,0)--cycle, gray); [/asy]$

We can also shift this quadrilateral one unit up, because of $v$ . Thus, this is our figure:

$[asy] import graph; Label f; unitsize(0.7cm); xaxis(-5,5,Ticks(f, 5.0, 1.0)); yaxis(-5,5,Ticks(f, 5.0, 1.0)); draw((0,0)--(-3,4)); draw((2,0)--(-1,4)); filldraw((0,0)--(-3,4)--(-1,4)--(2,0)--cycle, gray); filldraw((0,1)--(-3,5)--(-2,5)--(1,1)--cycle, gray); draw((0,0)--(0,1),black+dashed); draw((2,0)--(2,1),black+dashed); draw((-3,4)--(-3,5),black+dashed); draw((-1,4)--(-1,5),black+dashed); [/asy]$

$[asy] import graph; Label f; unitsize(0.7cm); xaxis(-5,5,Ticks(f, 5.0, 1.0)); yaxis(-5,5,Ticks(f, 5.0, 1.0)); draw((0,0)--(-3,4)); draw((1,0)--(-2,4)); filldraw((0,0)--(2,0)--(2,1)--(-1,5)--(-3,5)--(-3,4)--cycle, gray); [/asy]$

The length of the boundary is simply $1+1+5+1+1+5$ ( $5$ can be obtained by Pythagorean theorem, since we have side lengths $3$ and $4$ .). This equals $\boxed{\textbf{(E) }16.}$

~Technodoggo

Page

Toolbox

Search

2023 AMC 10B Problems/Problem 24

Solution 1