Mathematics
Problem of The Day
2024
94

94

Margie's winning art design is shown. The smallest circle has radius 2 cm, with each successive circle's radius increasing by 2 cm.

Which of the following is closest to the percent of the design that is black?

A. 42 B. 44 C. 45 D. 46 E. 48

Answer

A. 42

SOLUTIOn

Let the smallest circle be 1, the second smallest circle be 2, the third smallest circle be 3, etc.

\[\begin{array}{c|cc} \text{circle \#} & \text{radius} & \text{area} \\ \hline 1 & 2 & 4\pi \\ 2 & 4 & 16\pi \\ 3 & 6 & 36\pi \\ 4 & 8 & 64\pi \\ 5 & 10 & 100\pi \\ 6 & 12 & 144\pi \end{array}\]

The entire circle's area is

$144\pi$

The area of the black regions is

$(100-64)\pi + (36-16)\pi + 4\pi = 60\pi$

The percentage of the design that is black is

$\frac{60\pi}{144\pi} = \frac{5}{12} \approx \boxed{\textbf{(A)}\ 42}$

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