2016 AMC 10A Problem 10

Question

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is

1

foot wide on all four sides. What is the length in feet of the inner rectangle?

1

2

4

6

8

Answer: B

Concepts:area arithmetic sequence

Difficulty rating: 1370

Small hint:

Let the inner rectangle have length

l

Big hint:

The three region areas are

l,2l+6,2l+14

Solution:

Let

l

be the length of the inner rectangle. Then the area of the inner rectangle is

l.

The area of the middle region is going to be

(l + 2) \cdot 3 - l = 2l + 6.

The area of the outer region is

(l + 4) \cdot 5 - (l + 2) \cdot 3 = 2l + 14.

We know that these

3

values form an arithmetic sequence. That means that

\begin{aligned} l + 2l + 14 &= 2(2l + 6) \\ 3l + 14 &= 4l + 12 \\ l &= 2\end{aligned}

Thus, the correct answer is B .