2003 AMC 12B Problem 3

Question

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?

108

115

132

144

156

Answer: A

Concepts:optimization area

Difficulty rating: 1170

Small hint:

Find the area of each of the five regions from the figure

Big hint:

To minimize the total, put the most expensive flower in the smallest region and so on

Solution:

The five regions have areas

4,

6,

15,

20,

and

21

square feet.

To minimize the cost, plant the most expensive flowers in the smallest regions. The least possible cost is

\begin{aligned} &3(4) + 2.5(6) + 2(15) \\ &\quad {}+ 1.5(20) + 1(21) = 108. \end{aligned}

Thus, the correct answer is A.