Difference between revisions of "2013 AMC 8 Problems/Problem 23"
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<math>r</math> is the radius. let’s use the full circumference formula for a circle, which is | <math>r</math> is the radius. let’s use the full circumference formula for a circle, which is | ||
<math>2\pi r</math>. Since the semicircle is half of a circle, its arc length is | <math>2\pi r</math>. Since the semicircle is half of a circle, its arc length is | ||
− | \ | + | <math>\pi r</math>, which was given as |
− | < | + | <math>8.5\pi</math>. Solving for |
− | < | + | <math>r</math>, we get |
− | < | + | <math>r=8.5</math> |
. Therefore, the diameter, which is | . Therefore, the diameter, which is | ||
− | < | + | <math>2r</math>, is |
− | < | + | <math>2</math> x <math>8.5=17</math> |
Then, the other steps to solve the problem will be the same as mentioned above by SavannahSolver | Then, the other steps to solve the problem will be the same as mentioned above by SavannahSolver | ||
− | the answer is < | + | the answer is <math>\boxed{\textbf{(B)}\ 7.5}</math> |
Revision as of 21:22, 9 August 2025
Contents
Problem
Angle of
is a right angle. The sides of
are the diameters of semicircles as shown. The area of the semicircle on
equals
, and the arc of the semicircle on
has length
. What is the radius of the semicircle on
?
Video Solution
https://youtu.be/crR3uNwKjk0 ~savannahsolver
Solution 1
If the semicircle on were a full circle, the area would be
.
, therefore the diameter of the first circle is
.
The arc of the largest semicircle is , so if it were a full circle, the circumference would be
. So the
.
By the Pythagorean theorem, the other side has length , so the radius is
~Edited by Theraccoon to correct typos.
Brief Explanation
SavannahSolver got a diameter of because the given arc length of the semicircle was
. The arc length of a semicircle can be calculated using the formula
, where
is the radius. let’s use the full circumference formula for a circle, which is
. Since the semicircle is half of a circle, its arc length is
, which was given as
. Solving for
, we get
. Therefore, the diameter, which is
, is
x
Then, the other steps to solve the problem will be the same as mentioned above by SavannahSolver
the answer is
. - TheNerdWhoIsNerdy.
Minor edits by -Coin1
Solution 2
We go as in Solution 1, finding the diameter of the circle on and
. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largest is
, and the middle one is
, so the radius is
.
~Note by Theraccoon: The person who posted this did not include their name.
Video Solution by OmegaLearn
https://youtu.be/abSgjn4Qs34?t=2584
~ pi_is_3.14
Answer (B) 7.5
~ Mia Wang the Author ~skibidi ~Coin1