Difference between revisions of "2017 AMC 10A Problems/Problem 3"

Revision as of 16:12, 31 May 2021

Problem

Tamara has three rows of two $6$ -feet by $2$ -feet flower beds in her garden. The beds are separated and also surrounded by $1$ -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?

$[asy] draw((0,0)--(0,10)--(15,10)--(15,0)--cycle); fill((0,0)--(0,10)--(15,10)--(15,0)--cycle, lightgray); draw((1,1)--(1,3)--(7,3)--(7,1)--cycle); fill((1,1)--(1,3)--(7,3)--(7,1)--cycle, white); draw((1,4)--(1,6)--(7,6)--(7,4)--cycle); fill((1,4)--(1,6)--(7,6)--(7,4)--cycle, white); draw((1,7)--(1,9)--(7,9)--(7,7)--cycle); fill((1,7)--(1,9)--(7,9)--(7,7)--cycle, white); draw((8,1)--(8,3)--(14,3)--(14,1)--cycle); fill((8,1)--(8,3)--(14,3)--(14,1)--cycle, white); draw((8,4)--(8,6)--(14,6)--(14,4)--cycle); fill((8,4)--(8,6)--(14,6)--(14,4)--cycle, white); draw((8,7)--(8,9)--(14,9)--(14,7)--cycle); fill((8,7)--(8,9)--(14,9)--(14,7)--cycle, white); defaultpen(fontsize(8, lineskip=1)); label("2", (1.2, 2)); label("6", (4, 1.2)); defaultpen(linewidth(.2)); draw((0,8)--(1,8), arrow=Arrows); draw((7,8)--(8,8), arrow=Arrows); draw((14,8)--(15,8), arrow=Arrows); draw((11,0)--(11,1), arrow=Arrows); draw((11,3)--(11,4), arrow=Arrows); draw((11,6)--(11,7), arrow=Arrows); label("1", (.5,7.8)); label("1", (7.5,7.8)); label("1", (14.5,7.8)); label("1", (10.8,.5)); label("1", (10.8,3.5)); label("1", (10.8,6.5)); [/asy]$

$\textbf{(A)}\ 72\qquad\textbf{(B)}\ 78\qquad\textbf{(C)}\ 90\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 150$

Solution

Finding the area of the shaded walkway can be achieved by computing the total area of Tamara's garden and then subtracting the combined area of her six flower beds.

Since the width of Tamara's garden contains three margins, the total width is $2\cdot 6+3\cdot 1 = 15$ feet.

Similarly, the height of Tamara's garden is $3\cdot 2+4\cdot 1 = 10$ feet.

Therefore, the total area of the garden is $15\cdot 10 =150$ square feet.

Finally, since the six flower beds each have an area of $2\cdot 6 = 12$ square feet, the area we seek is $150 - 6\cdot 12$ , and our answer is $\boxed{\textbf{(B)}\ 78}$

Alternate Solution

Find the area of the entire rectangle, and subtract the beds. To do this, the length of the garden is $2+2+2+1+1+1+1=10$ The width is $6+6+1+1+1=15$ Therefore, the area of the entire garden is $15*10=150$ Each flower bed is $6*2=12$ , so the combined area of the beds is $12*6=72$ So, the total area of the walkways is $150-72=$ \boxed{\textbf{(B)}\ 78}$

Video Solution

https://youtu.be/str7kmcRMY8

https://youtu.be/B7UuRDIfXkQ

~savannahsolver

@@ Line 51: / Line 51: @@
 Finally, since the six flower beds each have an area of <math>2\cdot 6 = 12</math> square feet, the area we seek is <math>150 - 6\cdot 12</math>, and our answer is <math> \boxed{\textbf{(B)}\ 78}</math>
+==Alternate Solution==
+Find the area of the entire rectangle, and subtract the beds.
+To do this, the length of the garden is <math> 2+2+2+1+1+1+1=10 </math>
+The width is <math> 6+6+1+1+1=15 </math>
+Therefore, the area of the entire garden is <math> 15*10=150 </math>
+Each flower bed is <math> 6*2=12 </math>, so the combined area of the beds is <math> 12*6=72 </math>
+So, the total area of the walkways is <math> 150-72= </math> \boxed{\textbf{(B)}\ 78}$
 ==Video Solution==

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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