Difference between revisions of "2013 AMC 8 Problems/Problem 23"
m (→Solution 1) |
Theraccoon (talk | contribs) (→Solution 2) |
||
| (12 intermediate revisions by 4 users not shown) | |||
| Line 20: | Line 20: | ||
<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9</math> | <math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9</math> | ||
| − | |||
| − | |||
| − | |||
==Video Solution== | ==Video Solution== | ||
| Line 33: | Line 30: | ||
The arc of the largest semicircle is <math>8.5 \pi</math>, so if it were a full circle, the circumference would be <math>17 \pi</math>. So the <math>\text{diameter}=17</math>. | The arc of the largest semicircle is <math>8.5 \pi</math>, so if it were a full circle, the circumference would be <math>17 \pi</math>. So the <math>\text{diameter}=17</math>. | ||
| + | |||
| + | |||
By the Pythagorean theorem, the other side has length <math>15</math>, so the radius is <math>\boxed{\textbf{(B)}\ 7.5}</math> | By the Pythagorean theorem, the other side has length <math>15</math>, so the radius is <math>\boxed{\textbf{(B)}\ 7.5}</math> | ||
| + | |||
| + | ~Edited by Theraccoon to correct typos. | ||
| + | |||
| + | ==Brief Explanation== | ||
| + | SavannahSolver got a diameter of 17 because the given arc length of the semicircle was | ||
| + | 8.5π. The arc length of a semicircle can be calculated using the formula | ||
| + | πr, where | ||
| + | r is the radius. let’s use the full circumference formula for a circle, which is | ||
| + | 2πr. Since the semicircle is half of a circle, its arc length is | ||
| + | πr, which was given as | ||
| + | 8.5π. Solving for | ||
| + | r, we get | ||
| + | 𝑟=8.5 | ||
| + | . Therefore, the diameter, which is | ||
| + | 2r, is | ||
| + | 2x8.5=17 | ||
| + | Then, the other steps to solve the problem will be the same as mentioned above by SavannahSolver | ||
| + | the answer is <math>\boxed{\textbf{(B)}\ 7.5}</math> | ||
| + | |||
| + | |||
| + | . - TheNerdWhoIsNerdy. | ||
==Solution 2== | ==Solution 2== | ||
| − | We go as in Solution 1, finding the diameter of the circle on AC and AB. 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 <math>\frac{289\pi}{8}</math>, and the middle one is <math>\frac{289\pi}{8}-\frac{64\pi}{8}=\frac{225\pi}{8}</math>, so the radius is <math>\frac{15}{2}=\boxed{\textbf{(B)}\ 7.5}</math>. | + | We go as in Solution 1, finding the diameter of the circle on <math>\overline{AC}</math> and <math>\overline{AB}</math>. 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 <math>\frac{289\pi}{8}</math>, and the middle one is <math>\frac{289\pi}{8}-\frac{64\pi}{8}=\frac{225\pi}{8}</math>, so the radius is <math>\frac{15}{2}=\boxed{\textbf{(B)}\ 7.5}</math>. |
| + | |||
| + | ~Note by Theraccoon: The person who posted this did not include their name. | ||
==Video Solution by OmegaLearn== | ==Video Solution by OmegaLearn== | ||
Latest revision as of 17:47, 27 September 2024
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 
?
![[asy] import graph; pair A,B,C; A=(0,8); B=(0,0); C=(15,0); draw((0,8)..(-4,4)..(0,0)--(0,8)); draw((0,0)..(7.5,-7.5)..(15,0)--(0,0)); real theta = aTan(8/15); draw(arc((15/2,4),17/2,-theta,180-theta)); draw((0,8)--(15,0)); dot(A); dot(B); dot(C); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SE);[/asy]](http://latex.artofproblemsolving.com/2/6/2/2624acb19756f1b697f71413e9840070908905da.png)
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 17 because the given arc length of the semicircle was
8.5π. The arc length of a semicircle can be calculated using the formula
πr, where
r is the radius. let’s use the full circumference formula for a circle, which is
2πr. Since the semicircle is half of a circle, its arc length is
πr, which was given as
8.5π. Solving for
r, we get
𝑟=8.5
. Therefore, the diameter, which is
2r, is
2x8.5=17
Then, the other steps to solve the problem will be the same as mentioned above by SavannahSolver
the answer is
. - TheNerdWhoIsNerdy.
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
See Also
| 2013 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
