Difference between revisions of "2010 AMC 8 Problems/Problem 8"
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==Solution== | ==Solution== | ||
| − | Because they are both moving in the same direction, Emily is riding relative to Emerson <math>12-8=4</math> mph. Now we can look at it as if Emerson is not moving at all[on his skateboard] and Emily is riding at <math>4</math> mph. It takes her | + | Because they are both moving in the same direction, Emily is riding relative to Emerson <math>12-8=4</math> mph. Now we can look at it as if Emerson is not moving at all [on his skateboard] and Emily is riding at <math>4</math> mph. It takes her |
<cmath>\frac12 \ \text{mile} \cdot \frac{1\ \text{hour}}{4\ \text{miles}} = \frac18\ \text{hour}</cmath> | <cmath>\frac12 \ \text{mile} \cdot \frac{1\ \text{hour}}{4\ \text{miles}} = \frac18\ \text{hour}</cmath> | ||
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to ride the <math>1/2</math> mile to reach him, and then the same amount of time to be <math>1/2</math> mile ahead of him. This totals to | to ride the <math>1/2</math> mile to reach him, and then the same amount of time to be <math>1/2</math> mile ahead of him. This totals to | ||
| − | <cmath>2 \cdot \frac18 \ \text{hour} \cdot \frac{60\ \text{minutes}}{1\ \text{hour}} = \boxed{\textbf{(D)}\ 15}\ \text{minutes}</cmath>. | + | <cmath>2 \cdot \frac18 \ \text{hour} \cdot \frac{60\ \text{minutes}}{1\ \text{hour}} = \boxed{\textbf{(D)}\ 15}\ \text{minutes}</cmath> |
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| + | ==Video Solution by @MathTalks== | ||
| + | https://youtu.be/RhyRqHMXvq0?si=m1R2q8UnLRD-KksT | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/CdsOlV37534 | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2010|num-b=7|num-a=9}} | {{AMC8 box|year=2010|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 10:26, 18 November 2024
Problem
As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction 
mile in front of her. After she passes him, she can see him in her rear mirror until he is mile behind her. Emily rides at a constant rate of 
miles per hour, and Emerson skates at a constant rate of 
miles per hour. For how many minutes can Emily see Emerson?
Solution
Because they are both moving in the same direction, Emily is riding relative to Emerson 
mph. Now we can look at it as if Emerson is not moving at all [on his skateboard] and Emily is riding at 
mph. It takes her
![\[\frac12 \ \text{mile} \cdot \frac{1\ \text{hour}}{4\ \text{miles}} = \frac18\ \text{hour}\]](http://latex.artofproblemsolving.com/5/c/3/5c3c6fda6760c790b8771e388c649c6eea3d0835.png)
to ride the mile to reach him, and then the same amount of time to be mile ahead of him. This totals to
![\[2 \cdot \frac18 \ \text{hour} \cdot \frac{60\ \text{minutes}}{1\ \text{hour}} = \boxed{\textbf{(D)}\ 15}\ \text{minutes}\]](http://latex.artofproblemsolving.com/e/1/9/e19d6d55213e448f3e688cf3838fbbdb65a8d60b.png)
Video Solution by @MathTalks
https://youtu.be/RhyRqHMXvq0?si=m1R2q8UnLRD-KksT
Video Solution by WhyMath
See Also
| 2010 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
