Difference between revisions of "2017 AMC 10A Problems/Problem 7"
Supermessi (talk | contribs) (Created page with "Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia heade...") |
|||
| Line 1: | Line 1: | ||
| + | ==Problem== | ||
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? | Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? | ||
| − | (A) 30% (B) 40% (C) 50% (D) 60% (E) 70% | + | |
| + | <math>\textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 40\%\qquad\textbf{(C)}\ 50\%\qquad\textbf{(D)}\ 60\%\qquad\textbf{(E)}\ 70\%</math> | ||
| + | ==Solution== | ||
| + | |||
| + | Let <math>j</math> represent how far Jerry walked, and <math>s</math> represent how far Sylvia walked. Since the field is a square, and Jerry walked two sides of it, while Silvia walked the diagonal, we can simply define the side of the square field to be one, and find the distances they walked. Since Jerry walked two sides, | ||
| + | <math>j = 2</math> | ||
| + | Since Silvia walked the diagonal, she walked the hypotenuse of a 45, 45, 90 triangle with leg length 1. Thus, | ||
| + | <math>s = \sqrt{2} = 1.414...</math> | ||
| + | We can then take | ||
| + | <math>\frac{j-s}{j} = \frac{\sqrt{2 - 1.4}}{2} = 0.3 = 30\%</math> | ||
| + | <math>\boxed{ \textbf{A}}</math>. | ||
Revision as of 14:45, 8 February 2017
Problem
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
Solution
Let 
represent how far Jerry walked, and 
represent how far Sylvia walked. Since the field is a square, and Jerry walked two sides of it, while Silvia walked the diagonal, we can simply define the side of the square field to be one, and find the distances they walked. Since Jerry walked two sides,
Since Silvia walked the diagonal, she walked the hypotenuse of a 45, 45, 90 triangle with leg length 1. Thus,
We can then take
.
