All the units are feet and seconds except for Charlie's speed which is 8 inches per second. We need the same units for each of them and there are 12 inches in a foot so Charlie's speed is 8/12 = 2/3 feet per second.
When they start how far apart are Charlie and Snoopy? How fast is the distance between them decreasing? At that rate how long will it take the distance between them to decrease to zero feet? At that time how far has Charlie run?
Marie wrote back
I know that I should convert the inches to feet. I WANT to write the problem in a time rate distance sketch. Also I want to write equations so that I'm able to solve when Charlie and Sally pass each other how far will they be from the starting line? How do I write the equation for Charlie Sally and Snoopy? I'm thinking slope intercept form. Please help.
The equation you need is
If you want to find when Charlie and Sally pass each other you want distance to be the distance between them when the race starts, rate to be the rate at which the distance between them is decreasing and then time will be the time until the distance is 0, that is the time when Sally passes Charlie.
Once you have this time you can use distance = time × rate again. This time the rate is the rate at which Sally runs, 5 ft per second, time is the time when she passes Charlie and then distance will be her distance from the starting line at that time.
Marie's mother wrote back
Hi again Marie and mom,
As you can tell by my earlier responses I wouldn't have solved the problem this way, in fact I wouldn't have asked the question this way. I don't like problems that tell you to use a standard equation for the solution. I see so many students who say they can't do a problem because they can't remember the formula. Solving a mathematics problem is not about remembering a formula it's using your ingenuity and experience gained from solving prrevious problems to devise a way to approach a new problem. Anyway this is how the question was stated so let's do what we were asked.
I agree with the formulas that Marie used
t is in seconds and d is in feet. For one minute substitute t = 60 seconds so at 1 minute
At the end of the race d is one quarter of a mile which is 5280/4 = 1320 ft so
You can solve these for t to find the time when each of them crossed the finish line. For the dog you will find that t is negative, That's because he is running in the wrong direction and crossed the finish line before Sally and Charlie even started.
I agree with Marie's calculations for when Sally and Charlie pass each other.
When will Sally and the dog pass each other and how far will they be from the starting line?
I agree with the other calculations up to the questions about the end of the race. The end is at one quarter of a mile which is 1320 feet from the starting line and Charlie passes Sally at 1143 feet from the starting line so Charlie wins the race.
Use Charlie's formula, d = 6t to find the time when d = 1320 feet. Use this time and Sally's formula to calculate how far she is from the starting line when Charlie finished. The margin of victory (distance) is the distance she is from the finish line when Charlie finished. The margin of victory (time) is the additional time it will take her to reach the finish line after Charlie has finished.
I hope this helps,
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