Difference between revisions of "2015 AMC 10A Problems/Problem 3"
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Ann made a <math>3</math>-step staircase using <math>18</math> toothpicks as shown in the figure. How many toothpicks does she need to add to complete a <math>5</math>-step staircase? | Ann made a <math>3</math>-step staircase using <math>18</math> toothpicks as shown in the figure. How many toothpicks does she need to add to complete a <math>5</math>-step staircase? | ||
− | <math>\textbf{(A)}\ 9\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D) | + | <math>\textbf{(A)}\ 9\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24</math> |
+ | |||
+ | <asy> | ||
+ | size(150); | ||
+ | defaultpen(linewidth(0.8)); | ||
+ | path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); | ||
+ | for(int i=0;i<=2;i=i+1) | ||
+ | { | ||
+ | for(int j=0;j<=3-i;j=j+1) | ||
+ | { | ||
+ | filldraw(shift((i,j))*h,black); | ||
+ | filldraw(shift((j,i))*v,black); | ||
+ | } | ||
+ | }</asy> | ||
==Solution== | ==Solution== | ||
− | We can see that a <math>1</math>-step staircase requires <math>4</math> toothpicks and a <math>2</math>-step staircase requires <math>10</math> toothpicks. Thus, to go from a <math>1</math>-step to <math>2</math>-step staircase, <math>6</math> additional toothpicks are needed and to go from a <math>2</math>-step to <math>3</math>-step staircase, <math>8</math> additional toothpicks are needed. Applying this pattern, to go from a <math>3</math>-step to <math>4</math>-step staircase, <math>10</math> additional toothpicks are needed and to go from a <math>4</math>-step to <math>5</math>-step staircase, <math>12</math> additional toothpicks are needed. Our answer is <math>10+12=\boxed{\textbf{(D)}\ 22}</math>. | + | We can see that a <math>1</math>-step staircase requires <math>4</math> toothpicks and a <math>2</math>-step staircase requires <math>10</math> toothpicks. Thus, to go from a <math>1</math>-step to <math>2</math>-step staircase, <math>6</math> additional toothpicks are needed and to go from a <math>2</math>-step to <math>3</math>-step staircase, <math>8</math> additional toothpicks are needed. Applying this pattern, to go from a <math>3</math>-step to <math>4</math>-step staircase, <math>10</math> additional toothpicks are needed and to go from a <math>4</math>-step to <math>5</math>-step staircase, <math>12</math> additional toothpicks are needed. Our answer is <math>10+12=\boxed{\textbf{(D)}\ 22}</math> |
+ | |||
+ | ==Solution 2== | ||
+ | Alternatively, we can see with the <math>3</math>-step staircase has <math>2[2(3)+2+1]=18</math> toothpicks. Generalizing, we see that a staircase with <math>x</math> steps has <math>2[2x+(x-1)+(x-2)+...+1]</math> toothpicks. So, for <math>x=5</math> steps, we have <math>2[2(5)+4+3+2+1]=40</math> toothpicks. So our answer is <math>40-18=22</math> or <math>D</math>. | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/9BinymGHcUI | ||
+ | |||
+ | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2015|ab=A|num-b=2|num-a=4}} | {{AMC10 box|year=2015|ab=A|num-b=2|num-a=4}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 18:02, 16 June 2020
Problem
Ann made a -step staircase using toothpicks as shown in the figure. How many toothpicks does she need to add to complete a -step staircase?
Solution
We can see that a -step staircase requires toothpicks and a -step staircase requires toothpicks. Thus, to go from a -step to -step staircase, additional toothpicks are needed and to go from a -step to -step staircase, additional toothpicks are needed. Applying this pattern, to go from a -step to -step staircase, additional toothpicks are needed and to go from a -step to -step staircase, additional toothpicks are needed. Our answer is
Solution 2
Alternatively, we can see with the -step staircase has toothpicks. Generalizing, we see that a staircase with steps has toothpicks. So, for steps, we have toothpicks. So our answer is or .
Video Solution
~savannahsolver
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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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