2021 AMC 10B Problems/Problem 11
Contents
Problem
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?
Solution 1:
Let the side lengths of this rectangular pan be 
and 
; it follows that 
. This gives 
after some manipulation, so 
. By inspection, 
maximizes the number of brownies ~ ike.chen
Solution 2:
Let the dimensions of the rectangular pan be 
and 
. The number of interior pieces is 
because you cannot include the border, and the number of pieces along the perimeter is 
(THIS PART IS FLAWED ~ anonymous user).
Setting these two expressions equal, we have 
Applying SFFT (Simon's Favorite Factoring Trick), we get 
. Doing a bit of trial-and-error, we see that 
is maximum when 
and 
, which gives us a maximum of 
brownies. 
.
~Bryguy
Video Solution by OmegaLearn (Simon's Favorite Factoring Trick)
~ pi_is_3.14
See Also
| 2021 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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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