Difference between revisions of "2006 AMC 10A Problems/Problem 12"
m |
Dairyqueenxd (talk | contribs) (→Solution) |
||
(9 intermediate revisions by 5 users not shown) | |||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
− | + | <!-- [[Image:2006 AMC 10A-12.GIF]] --> | |
+ | |||
+ | Rolly wishes to secure his dog with an 8-foot rope to a [[square (geometry) | square]] shed that is 16 feet on each side. His preliminary drawings are shown. | ||
− | + | <asy> | |
+ | size(150); pathpen = linewidth(0.6); defaultpen(fontsize(10)); | ||
+ | D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle); | ||
+ | D((16,-8)--(24,-8)); | ||
+ | label('Dog', (24, -8), SE); | ||
+ | MP('I', (8,-8), (0,0)); | ||
+ | MP('8', (16,-4), W); | ||
+ | MP('8', (16,-12),W); | ||
+ | MP('8', (20,-8), N); | ||
+ | label('Rope', (20,-8),S); | ||
+ | D((0,-20)--(16,-20)--(16,-36)--(0,-36)--cycle); | ||
+ | D((16,-24)--(24,-24)); | ||
+ | MP("II", (8,-28), (0,0)); | ||
+ | MP('4', (16,-22), W); | ||
+ | MP('8', (20,-24), N); | ||
+ | label("Dog",(24,-24),SE); | ||
+ | label("Rope", (20,-24), S); | ||
+ | </asy> | ||
Which of these arrangements give the dog the greater [[area]] to roam, and by how many square feet? | Which of these arrangements give the dog the greater [[area]] to roam, and by how many square feet? | ||
− | <math> \ | + | <math> \textbf{(A) } I,\,\textrm{ by }\,8\pi\qquad \textbf{(B) } I,\,\textrm{ by }\,6\pi\qquad \textbf{(C) } II,\,\textrm{ by }\,4\pi\qquad \textbf{(D) } II,\,\textrm{ by }\,8\pi\qquad \textbf{(E) } II,\,\textrm{ by }\,10\pi </math> |
+ | |||
== Solution == | == Solution == | ||
− | Let us first examine the area of both possible arrangements. The rope outlines a circular boundary that the dog may dwell in. Arrangement I allows the dog | + | <!-- [[Image:2006 AMC 10A-12b.GIF]] --> |
− | <math>\frac12\cdot(\pi\cdot8^2) = 32\pi</math> square feet of area. Arrangement II allows <math>32\pi</math> square feet plus a little more on the top part of the fence. So we already know that Arrangement II allows more freedom - only thing left is to find out how much. The extra area can be represented by a quarter of a circle with radius 4. So the extra area is | + | <asy> |
− | <math>\frac14\cdot(\pi\cdot4^2) = 4\pi</math>. Thus the answer is <math>\ | + | size(150); pathpen = linewidth(0.7); defaultpen(linewidth(0.7)+fontsize(10)); |
− | + | filldraw(arc((16,-8),8,-90,90)--cycle, rgb(0.9,0.9,0.6)); | |
− | + | filldraw(arc((16,-26),8,-90,90)--cycle, rgb(0.9,0.9,0.6)); | |
− | + | filldraw(arc((16,-22),4,90,180)--(16,-22)--cycle, rgb(0.9,0.9,0.6)); | |
− | + | D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle); | |
− | + | D((16,-8)--(24,-8)); | |
+ | label('Dog', (24, -8), SE); | ||
+ | MP('I', (8,-8), (0,0)); | ||
+ | MP('8', (16,-4), W); | ||
+ | MP('8', (16,-12),W); | ||
+ | MP('8', (20,-8), N); | ||
+ | label('Rope', (20,-8),S); | ||
+ | pair sD = (0,-2); | ||
+ | D(shift(sD)*((0,-20)--(16,-20)--(16,-36)--(0,-36)--cycle)); | ||
+ | D(shift(sD)*((16,-24)--(24,-24))); | ||
+ | MP("II", (8,-28)+sD, (0,0)); | ||
+ | MP('4', (16,-22)+sD, W); | ||
+ | MP('8', (20,-24)+sD, N); | ||
+ | label("Dog",(24,-24)+sD,SE); | ||
+ | label("Rope", (20,-24)+sD, S); | ||
+ | </asy> | ||
+ | Let us first examine the area of both possible arrangements. The rope outlines a circular boundary that the dog may dwell in. Arrangement <math>I</math> allows the dog | ||
+ | <math>\frac12\cdot(\pi\cdot8^2) = 32\pi</math> square feet of area. Arrangement <math>II</math> allows <math>32\pi</math> square feet plus a little more on the top part of the fence. So we already know that Arrangement <math>II</math> allows more freedom - only thing left is to find out how much. The extra area can be represented by a quarter of a circle with radius 4. So the extra area is | ||
+ | <math>\frac14\cdot(\pi\cdot4^2) = 4\pi</math>. Thus the answer is <math>\boxed{\textbf{(C) } II,\,\textrm{ by }\,4\pi}</math>. | ||
− | + | == See also == | |
+ | {{AMC10 box|year=2006|ab=A|num-b=11|num-a=13}} | ||
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
+ | [[Category:Area Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 09:14, 17 December 2021
Problem
Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown.
Which of these arrangements give the dog the greater area to roam, and by how many square feet?
Solution
Let us first examine the area of both possible arrangements. The rope outlines a circular boundary that the dog may dwell in. Arrangement allows the dog square feet of area. Arrangement allows square feet plus a little more on the top part of the fence. So we already know that Arrangement allows more freedom - only thing left is to find out how much. The extra area can be represented by a quarter of a circle with radius 4. So the extra area is . Thus the answer is .
See also
2006 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.