   SEARCH HOME Math Central Quandaries & Queries  Question from Marie, a student: Sally and Charlie are having a race. Charlie gets a 900 foot lead and runs 8 inches per second. Sally begins at the starting line and runs at a rate of 5 ft per second. Charlie's dog is also in the race. Snoopy starts 1100 feet ahead of Sally and runs toward the starting line at a rate of 1 foot per second. When will Charlie and Snoopy past each other and how far will they be from the starting line? Hi Marie,

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?

Penny

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.

Marie,

The equation you need is

distance = time × rate

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.

Penny

Marie's mother wrote back

Hello Penny:

Will you please give my high school student some assistance on the following distance problem?

Sally and Charlie are having a race. Sally gets a 1,000 foot lead and runs at 9 inches per second. Charlie begins at the starting line and runs at a rate of 6 feet per second. Their dog starts 1,200 feet ahead of of Charlie and runs back towards the starting line at a rate of 2 feet per second.

Write equations for each of the runners, relating time t to distance from the starting line d. Also, include an equation for both the one minute mark and the finish line.

Are these equations correct?

Sally: d = 3/4t + 1000

Charlie: d= 6t

Dog: d=-2t + 1200

One minute mark: d= t(60)

Finish line d= t(5280)

When will Sally and Charlie pass each other and how far will they be from the starting line?

Sally

(4)y = 3/4x + 1000 (4)
4y = 3x + 4000

-3x + 4y = 4000
-3x = 4(6x) = 4000
-3x + 24x =4000

21x/21=4000/21
x= 190.4

y=6x
y=6(190.4)
y= 1142.8

Time 190.4 seconds

Distance from Start 1142.8 feet

When will Sally and the dog pass each other and how far will they be from the starting line?

I'm not able to calculate

Time: seconds

Distance from start: feet

When will the Dog and Charlie pass each other and how far will they be from the starting line?

Time: 150 seconds

Distance from start 900 feet

After one minute into the race, how far will each runner be?

Sally 1045 Feet
Charlie 360 Feet
Dog 1080 Feet

When will the dog cross the starting line?

Time: 6000 seconds

If the race is a quarter mile lon, who will win and what will be the margin of victory (both time and distance)?

Winner. Sally

Margin of Victory 426.6

1380 feet

Penny I did not bother to type all of my high school students calculations after the first question. I just typed her answers. Please assist us with this. Your help is greatly appreciated.

Thank you

-Marie's Mother

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

Sally: d = 3/4t + 1000

Charlie: d= 6t

Dog: d=-2t + 1200

t is in seconds and d is in feet. For one minute substitute t = 60 seconds so at 1 minute

Sally: d = (3/4)(60) + 1000 = 1045 ft
Charlie: d = 6(60) = 360 ft
Dog: d = -2(60) + 1200 = 1080 ft

At the end of the race d is one quarter of a mile which is 5280/4 = 1320 ft so

Sally: 1320 = 3/4 t + 1000 ft
Charlie: 1320 = 6 t ft
Dog: 1320 = -2 t + 1200 ft

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?

When Sally and Snoopy pass each other they are the same distance from the starting line. If the time when they meet is t seconds then Sally is 3/4 t + 1000 feet from the starting line and Snoopy is -2t + 1200 feet from the starting line thus

3/4 t + 1000 = -2 t + 1200

multiply both sides by 4

3 t + 4000 = -8 t + 4800
11 t = 800
t = 800/11 = 72.7 seconds

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,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.