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Difference between revisions of "2003 AMC 10B Problems/Problem 4"

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==Problem 4==
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Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost <math> \ </math><math>1</math> each, begonias <math> \ </math><math>1.50</math> each, cannas <math> \ </math><math>2</math> each, dahlias <math> \ </math><math>2.50</math> each, and Easter lilies <math> \ </math><math>3</math> each. What is the least possible cost, in dollars, for her garden?
 
Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost <math> \ </math><math>1</math> each, begonias <math> \ </math><math>1.50</math> each, cannas <math> \ </math><math>2</math> each, dahlias <math> \ </math><math>2.50</math> each, and Easter lilies <math> \ </math><math>3</math> each. What is the least possible cost, in dollars, for her garden?

Revision as of 19:45, 22 June 2011

Problem

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $$$1$ each, begonias $$$1.50$ each, cannas $$$2$ each, dahlias $$$2.50$ each, and Easter lilies $$$3$ each. What is the least possible cost, in dollars, for her garden?

[asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((6,0)--(0,0)--(0,1)--(6,1)); draw((0,1)--(0,6)--(4,6)--(4,1)); draw((4,6)--(11,6)--(11,3)--(4,3)); draw((11,3)--(11,0)--(6,0)--(6,3)); label("1",(0,0.5),W); label("5",(0,3.5),W); label("3",(11,1.5),E); label("3",(11,4.5),E); label("4",(2,6),N); label("7",(7.5,6),N); label("6",(3,0),S); label("5",(8.5,0),S);[/asy]

$\textbf{(A) } 108 \qquad\textbf{(B) } 115 \qquad\textbf{(C) } 132 \qquad\textbf{(D) } 144 \qquad\textbf{(E) } 156$

Solution

The areas of the five regions from greatest to least are $21,20,15,6$ and $4$.

If we want to minimize the cost, we want to maximize the area of the cheapest flower and minimize the area of the most expensive flower. Doing this, the cost is $1\cdot21+1.50\cdot20+2\cdot15+2.50\cdot6+3\cdot4$, which simplifies to $$$108$. Therefore the answer is $\boxed{\textbf{(A) } 108}$.

See Also

2003 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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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