AMC 8 Daily Practice Round 7

Complete problem set with solutions and individual problem pages

Problem 23 Easy

In the figure on the right, two sectors with central angles of 90^\circ are placed on top of a large circle, and a small circle is placed on top of the two sectors. All four shapes share the same center. If the ratio of the radii of the small circle, the large circle, and the sectors is 1 : 3 : 4, what percentage of the total area is occupied by the shaded region?

  • A.

    32\%

  • B.

    20\%

  • C.

    28\%

  • D.

    36\%

  • E.

    40\%

Answer:A

Let the radii of the large circle, the small circle, and the sectors be r, 3r, and 4r respectively.

 

The total area of the logo is

\frac{1}{2}\pi (4r)^2 + \frac{1}{2}\pi (3r)^2 = 12.5\pi r^2.

 

The area of the shaded region is

\frac{1}{2}\pi (3r)^2 - \frac{1}{2}\pi (r)^2 = 4\pi r^2.

 

Therefore, the shaded region occupies

\frac{4\pi r^2}{12.5\pi r^2} = 32\%

of the total area of the logo.

Related Problems
Problem 21 AMC 8 Daily Practice Round 7
Problem 22 AMC 8 Daily Practice Round 7
Problem 24 AMC 8 Daily Practice Round 7
Problem 25 AMC 8 Daily Practice Round 7
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