Difference between revisions of "2011 AMC 10B Problems/Problem 8"

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== Problem 8 ==
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== Problem==
  
 
At a certain beach if it is at least <math>80^{\circ} F</math> and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?
 
At a certain beach if it is at least <math>80^{\circ} F</math> and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?
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==Solution==
 
==Solution==
  
<math>C, D,</math> and <math>E</math> can not be right because the sun intensity of June 10 does not depend at all on the temperature.
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The beach not being crowded only means that it is not both hot and sunny. Equivalently, it is either cool or cloudy, which means <math>\boxed{\textbf{(B)}}</math> is correct.
  
If it was cooler than <math>80^\circ</math> but it was still sunny, it would not meet the conditions of <math>A,</math> but the beach would still be not crowded.
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It could be in fact cool and sunny, which would violate '''(A)''' and '''(E)'''. It could also be cool and cloudy, which would violate '''(D)''', or hot and cloudy, which would violate '''(C)'''.
 
 
However, <math>B</math> means that if either condition was not met, then the beach would not be crowded. Therefore, the correct answer is <math>\boxed{\textbf{(B)}}</math>
 
  
 
== See Also==
 
== See Also==
  
 
{{AMC10 box|year=2011|ab=B|num-b=7|num-a=9}}
 
{{AMC10 box|year=2011|ab=B|num-b=7|num-a=9}}
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{{MAA Notice}}

Latest revision as of 21:04, 5 February 2018

Problem

At a certain beach if it is at least $80^{\circ} F$ and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?

$\textbf{(A)}\ \text{The temperature was cooler than } 80^{\circ} \text{F and it was not sunny.}$

$\textbf{(B)}\ \text{The temperature was cooler than } 80^{\circ} \text{F or it was not sunny.}$

$\textbf{(C)}\ \text{If the temperature was at least } 80^{\circ} \text{F, then it was sunny.}$

$\textbf{(D)}\ \text{If the temperature was cooler than } 80^{\circ} \text{F, then it was sunny.}$

$\textbf{(E)}\ \text{If the temperature was cooler than } 80^{\circ} \text{F, then it was not sunny.}$

Solution

The beach not being crowded only means that it is not both hot and sunny. Equivalently, it is either cool or cloudy, which means $\boxed{\textbf{(B)}}$ is correct.

It could be in fact cool and sunny, which would violate (A) and (E). It could also be cool and cloudy, which would violate (D), or hot and cloudy, which would violate (C).

See Also

2011 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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

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