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AMC10每日一题（2000年真题#18）

2018-04-27
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2000 AMC 10 竞赛试题/第18题

Problem

Charlyn walks completely around the boundary of a square whose sides are each $5$ km long. From any point on her path she can see exactly $1$ km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?

$\mathrm{(A)} 24 \qquad\mathrm{(B)}\ 27 \qquad\mathrm{(C)}\ 39 \qquad\mathrm{(D)}\ 40 \qquad\mathrm{(E)}\ 42$

Solution

The area she sees looks at follows:

AMC10

The part inside the walk has area $5\cdot 5 - 3\cdot 3 = 16$ . The part outside the walk consists of four rectangles, and four arcs. Each of the rectangles has area $5\cdot 1=5$ . The four arcs together form a circle with radius $1$ .

Therefore the total area she can see is $16 + 4\cdot 5 + \pi\cdot 1^2 = 36+\pi \simeq 39.14$ , which rounded to the nearest integer is $39$ . $\boxed{C}$