Difference between revisions of "2003 AMC 10B Problems/Problem 16"
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− | {{ | + | ==Problem== |
+ | |||
+ | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that a restaurant should offer so that a customer could have a different dinner each night in the year <math>2003</math>?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> | ||
+ | |||
+ | <math>\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 7 \qquad\textbf{(E) } 8</math> | ||
+ | |||
+ | ==Solution== | ||
+ | |||
+ | Let <math>m</math> be the number of main courses the restaurant serves, so <math>2m</math> is the number of appetizers. Then the number of dinner combinations is <math>2m\times m\times3=6m^2</math>. Since the customer wants to eat a different dinner in all <math>365</math> days of <math>2003</math>, we must have | ||
+ | |||
+ | <cmath>\begin{align*} | ||
+ | 6m^2 &\geq 365\ | ||
+ | m^2 &\geq 60.83\ldots.\end{align*}</cmath> | ||
+ | |||
+ | The smallest integer value that satisfies this is <math>\boxed{\textbf{(E)}\ 8}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2003|ab=B|num-b=15|num-a=17}} | ||
+ | {{MAA Notice}} |
Revision as of 09:26, 4 March 2018
Problem
A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that a restaurant should offer so that a customer could have a different dinner each night in the year ?
Solution
Let be the number of main courses the restaurant serves, so is the number of appetizers. Then the number of dinner combinations is . Since the customer wants to eat a different dinner in all days of , we must have
The smallest integer value that satisfies this is .
See Also
2003 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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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