1962 AMC 12 Problem 16

Attempt Problem 16 of the 1962 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1962 AMC 12 solutions, or check the answer key.

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16.

Given rectangle

R_1

with one side

2

inches and area

12

square inches. Rectangle

R_2

with diagonal

15

inches is similar to

R_1.

Expressed in square inches the area of

R_2

is:

\dfrac92

36

\dfrac{135}{2}

9\sqrt{10}

\dfrac{27\sqrt{10}}4

The sides of

R_1

are

2

and

6

Areas scale as the square of the ratio of corresponding diagonals

The diagonal of

R_1

\sqrt{2^2+6^2}=2\sqrt{10}.

Therefore

\frac{[R_2]}{[R_1]} =\left(\frac{15}{2\sqrt{10}}\right)^2 =\frac{45}{8}.

Thus

[R_2]=12(\frac{45}{8})=\frac{135}{2}.

Therefore, the correct answer is C.

Problem 16 in other years