no id - Chat started on: Nov 4, 2007 10:25:11 PM [10:25 PM] Corey Couillard has entered the room [10:25 PM] Kate Sims has entered the room [10:25 PM] This room is not anonymous. [10:25 PM] ziizoo admin: Session started on Sun Nov 04 23:22:14 EST 2007 [10:25 PM] Kate Sims: Hi Corey [10:25 PM] Corey Couillard: I have a word problem that I need to solve and graph. should i give you the problem [10:25 PM] Kate Sims: Yes, just type it in [10:28 PM] Corey Couillard: Rematch of tortoise and the hare. The tortoise gets a 1000ft lead and runs at a rate of 9 inches per second. The hare begins at the starting point and runs at 6 ft per second. The race is a quarter mile long. The first question is when will the tortoise and hare pass each and how far will they be from the starting line [10:29 PM] Kate Sims: Okay, I always start problems like this by making a rate-time-distance chart. Let's go to the whiteboard... [10:29 PM] Kate Sims: Are you able to see the boxes I just drew? [10:32 PM] Corey Couillard: yes, guideline was that all unit of measures must be equal so I converted to ft [10:32 PM] Kate Sims: So then how fast is the tortoise going? [10:33 PM] Corey Couillard: 3/4ft per sec [10:33 PM] Kate Sims: Good, let's fill taht in [10:33 PM] Kate Sims: Are you able to draw on the same whiteboard as me? [10:34 PM] Corey Couillard: yes [10:34 PM] Kate Sims: Can you fill in the hare's rate? [10:35 PM] Kate Sims: Okay, what about the times [10:36 PM] Corey Couillard: was per second but didn't know if I should convert to minutes as I need to graph [10:36 PM] Kate Sims: Let me think for a second about how it's going to turn out... [10:37 PM] Kate Sims: Yes, let's put it into ft/min [10:38 PM] Corey Couillard: ok than 3/4 needs to change to 45 [10:38 PM] Corey Couillard: and 6 to 360 [10:39 PM] Kate Sims: Okay, good. I'm having trouble deleting, so I just added it into the box. [10:39 PM] Kate Sims: Let's say that the tortoise has been running for t minutes [10:39 PM] Kate Sims: How long has the hare been running for? [10:40 PM] Corey Couillard: t minutes [10:40 PM] Kate Sims: Good [10:40 PM] Kate Sims: Okay, now that we have rate and time, what is each's distance? [10:41 PM] Corey Couillard: tort 1045 after 1 min, hare 360ft after 1 min [10:41 PM] Kate Sims: What about after t minutes? [10:42 PM] Corey Couillard: i did a table up to 1320 for each , 8 minuts for tort to reach 1360 and 4 minutes for hare to reach 1440 [10:43 PM] Kate Sims: So what are you doing each time? If I said 5 minutes, what would you do? [10:44 PM] Kate Sims: Hard to read... (I'm having trouble too). Why don't you type it? [10:46 PM] Corey Couillard: i used y=dt+b so, y=45(5) +1000, y=1225 [10:47 PM] Kate Sims: Oh, I see, good. Now instead of 5, we're using t, but everything else is exactly the same... [10:48 PM] Corey Couillard: not sure where they intercept....do I need to graph to determine [10:49 PM] Kate Sims: Yes, you'll need to graph, but what's your equation for after time t? [10:49 PM] Corey Couillard: not sure what you are asking [10:49 PM] Kate Sims: You said that if you plug in 5, the tortoise's distance is 45(5) + 1000 [10:49 PM] Kate Sims: What if I said instead of 5, just do for any time t [10:50 PM] Corey Couillard: y=45t+1000 [10:50 PM] Kate Sims: Exactly [10:51 PM] Kate Sims: And the equation for the hare? [10:52 PM] Kate Sims: Great. Are you okay on graphing those 2 lines using the y-int and the slope? [10:53 PM] Corey Couillard: no, don't know how to grahp [10:54 PM] Kate Sims: Okay, so what I'd do is make sure you have enough space for them to finish the race. Since y represents your distance, it should go up to the distance of the race. How many feet is the race? [10:54 PM] Corey Couillard: 1320 ft [10:55 PM] Kate Sims: Okay, so I'd use a whole piece of graph paper, and then figure out how many feet every box should be so that it fits... [10:55 PM] Kate Sims: Use the long side for y, and put your axes along the left and bottom edges (you don't need any negative times or distances!) [10:55 PM] Kate Sims: I'm going to type the equations here and then clear the whiteboard. [10:56 PM] Kate Sims: Tortoise: y = 45t + 1000; Hare = 360t [10:57 PM] Kate Sims: What do you think? How many feet per box? [10:58 PM] Corey Couillard: 100 [10:58 PM] Kate Sims: That'll work, or if you have 26 boxes, you can do 50 per box and be a little more accurate [11:01 PM] Kate Sims: Okay, are you ready to graph? [11:02 PM] Corey Couillard: yes [11:02 PM] Kate Sims: Let's do the tortoise. Start at the y-intercept and make a dot on the graph... [11:03 PM] Kate Sims: What is the slope written as a fraction? [11:03 PM] Corey Couillard: 45/1 [11:04 PM] Kate Sims: Good, so let's go up 45 and right 1 from your dot and make another point [11:04 PM] Kate Sims: This is going to be imperfect, and in fact the answer is a messy decimal, so they can't expect us to get an exact answer by graphing it, but we'll get a rough estimate... [11:05 PM] Kate Sims: Now connect the two... [11:05 PM] Kate Sims: Why don't you hit undo and I'll show you the line tool [11:06 PM] Kate Sims: See the tool next to the pencil? It'll make a nice straight line...exactly [11:06 PM] Kate Sims: In fact, make it longer so that it reaches the top of the graph (where the finish is) [11:06 PM] Kate Sims: Then graph the hare's line the same way [11:08 PM] Kate Sims: Good. So after about how many minutes do they meet? [11:09 PM] Corey Couillard: just over 3 minutes [11:09 PM] Kate Sims: Yep. And how far away are they from the starting line? [11:10 PM] Corey Couillard: approx 1145ft [11:10 PM] Kate Sims: Did you get that by plugging in? I can't imagine you got that exact a number by eyeballing it! [11:11 PM] Corey Couillard: i eye balled it....is there a way to get the exact number [11:11 PM] Kate Sims: Or is your graph just much better than my crappy one :) [11:11 PM] Kate Sims: Yes, there's a much better way than graphing it. [11:12 PM] Corey Couillard: can you give me the equation [11:12 PM] Kate Sims: Our two equations are y=360t and y=45t + 1000. How would you find when they cross? [11:12 PM] Corey Couillard: times t until equal [11:12 PM] Kate Sims: In other words, what do you set equal? [11:13 PM] Corey Couillard: i don't know [11:13 PM] Corey Couillard: i have the table completed and only see numbers that are close [11:13 PM] Kate Sims: So we're looking for when their distances are =. What variable stands for distance? [11:14 PM] Corey Couillard: 3 minutes [11:14 PM] Corey Couillard: d [11:14 PM] Kate Sims: Yes, it's close to 3. But to do it exact, which is distance, "y" or "t"? [11:14 PM] Corey Couillard: y [11:15 PM] Kate Sims: Good, so if we know that the y's are = and that y = 360t and y = 45t + 1000, then that means that 360t = 45t + 1000 [11:16 PM] Kate Sims: Solve for t... [11:19 PM] Kate Sims: Still thinking? [11:19 PM] Corey Couillard: i don't know how to do [11:19 PM] Kate Sims: What would you do to both sides to get the t's together? [11:20 PM] Corey Couillard: subtract 45t from both sides [11:20 PM] Kate Sims: Good, do that, then you're practically done! [11:20 PM] Kate Sims: What's the next step? [11:20 PM] Corey Couillard: then do I sub 315 from each side [11:21 PM] Kate Sims: Ahh. but 315t means 315 times t. What do you do to get rid of the 315 [11:21 PM] Corey Couillard: divide 315 by both sides [11:21 PM] Kate Sims: Exactly. Then you could just reduce 1000/315, since it doesn't go in evenly... [11:22 PM] Kate Sims: It should be 200/63 also known as 3 11/63 minutes--yuck! [11:23 PM] Kate Sims: Once you know the time "t", how do you get their distances? [11:25 PM] Corey Couillard: don't know [11:26 PM] Kate Sims: What did we say represents distance? [11:26 PM] Corey Couillard: y [11:26 PM] Kate Sims: Good, so plug into one of the equations. Since they're in the same place, you should get the same answer no matter where you plug in (I'd pick the hare) [11:27 PM] Kate Sims: Once again it's ugly, so reduce and then just stop [11:28 PM] Kate Sims: Don't forget to cross-cancel before you multiply... [11:29 PM] Kate Sims: How's it going? I wrote the beginning on the whiteboard... [11:29 PM] Corey Couillard: 8000/7 [11:30 PM] Kate Sims: Yep. Try dividing on a calculator to see how close you were with your estimate [11:30 PM] Kate Sims: I couldn't believe how close you got!!! [11:31 PM] Kate Sims: Are there still more parts to the problem? [11:32 PM] Corey Couillard: 1142.85 [11:32 PM] Kate Sims: Yes, so you were only off by just over 2 feet. Very impressive. [11:33 PM] Kate Sims: Does the problem want anything else? [11:34 PM] Corey Couillard: another question: Equations for each runner relating to time t from the starting point d. [11:34 PM] Corey Couillard: is that what we already did y=dt+b [11:35 PM] Kate Sims: Yes, that sounds like our equations. [11:35 PM] Corey Couillard: also asking for equations for runners relating time and distance from the finish line [11:36 PM] Kate Sims: Here's an easy way... Say I've run 300 ft. How many feet am I from the finish line? [11:36 PM] Corey Couillard: 1020ft [11:36 PM] Kate Sims: Good. So if the hare has run 360t, then how far from the finish line is he? [11:37 PM] Corey Couillard: 3.66666 [11:37 PM] Kate Sims: Remember, now we're doing this in general. How did you get the 1020? [11:38 PM] Corey Couillard: 360t-1320 [11:38 PM] Kate Sims: Almost. Think carefully [11:39 PM] Kate Sims: Do you see the mistake? If not, just say so. [11:39 PM] Corey Couillard: 1320=360t+b [11:40 PM] Kate Sims: No, you were closer with 360t - 1320. [11:40 PM] Corey Couillard: i don't know [11:40 PM] Kate Sims: Hint: If I've run 300 feet then I have 1320 - 300 to go [11:40 PM] Kate Sims: So if I've run 360t feet then I have.... [11:41 PM] Corey Couillard: ? [11:42 PM] Kate Sims: All we're doing is putting 360t in place of the number I made up (300) [11:44 PM] Kate Sims: I made a little chart on the whiteboard if you're still stuck... [11:45 PM] Corey Couillard: 1320/360t [11:45 PM] Kate Sims: What operation were we doing every time? [11:45 PM] Kate Sims: add/subtract/multiply/divide? [11:47 PM] Kate Sims: Let me know if you're still confused. [11:47 PM] Corey Couillard: still confused [11:48 PM] Kate Sims: If I've run 0 miles then I have 1320 - 0 to go [11:48 PM] Kate Sims: If I've run 300 miles then I have 1320 - 300 to go [11:48 PM] Kate Sims: What am I doing every time? [11:48 PM] Corey Couillard: subtracting [11:48 PM] Kate Sims: Yes, subtracting from 1320. [11:49 PM] Kate Sims: So if I've gone 360t miles then I have ________ miles left to go [11:49 PM] Corey Couillard: 960mi [11:49 PM] Kate Sims: Don't forget there's a "t" there so I can't actually subtract [11:51 PM] Kate Sims: Just write out the subtraction and stop [11:51 PM] Corey Couillard: 1320-360t [11:51 PM] Kate Sims: Good, so for the hare the distance to finish (let's call it f) is f = 1320 - 360t [11:52 PM] Kate Sims: Now for the tortoise... If he's gone 45t + 1000 feet, how would I get the number of feet he has left? [11:53 PM] Corey Couillard: f=1000-45t [11:53 PM] Kate Sims: But wait, if you know the number of feet he's gone, what do you do with it? [11:54 PM] Kate Sims: If you're not sure, scroll back up to 11:48pm! [11:56 PM] Kate Sims: Stuck? I've updated our whiteboard a little [11:56 PM] Corey Couillard: f=1045-45t [11:57 PM] Kate Sims: Close. Why 1045? [11:57 PM] Corey Couillard: f=1320-1045t [11:58 PM] Kate Sims: Hmmm, you're close but not there. I agree it starts f = 1320 - _____________ [11:58 PM] Kate Sims: What should go in the blank? [12:00 AM] Kate Sims: I put a hint on the whiteboard [12:01 AM] Corey Couillard: -1045t [12:01 AM] Kate Sims: Explain... [12:02 AM] Corey Couillard: starting point 1000 increased by 45ft per minute [12:02 AM] Kate Sims: Okay, but your'e assuming it decreases by 1045 every minute. It only decreases by the 1000 once (at the very beginning). [12:03 AM] Kate Sims: Try actually doing less thinking... Look on the whiteboard and use the pattern. Don't combine anything. [12:05 AM] Corey Couillard: 1320-45t [12:05 AM] Kate Sims: But now you're leaving out the 1000! [12:05 AM] Corey Couillard: f=1320-45t+1000 [12:06 AM] Kate Sims: Yes, but you need parentheses so the whole thing is subtracted: f = 1320 - (45t + 1000) [12:06 AM] Kate Sims: Simplify that, and you have your answer! [12:08 AM] Corey Couillard: ? [12:08 AM] Kate Sims: How would you simplify the 1320 - (45t + 1000)? [12:08 AM] Corey Couillard: divide by 45 [12:09 AM] Kate Sims: No, nothing that fancy. You just want to get rid of the ( ). What do you do with the minus sign? [12:10 AM] Corey Couillard: i don't know [12:10 AM] Kate Sims: So you distribute the minus sign, as I showed with arrows on the whiteboard. What does that give you? [12:12 AM] Kate Sims: Stuck? [12:12 AM] Corey Couillard: yes [12:12 AM] Kate Sims: So you put the minus on each thing inside the parentheses to get 1320 - 45t - 1000 [12:13 AM] Kate Sims: Then combine like terms, and what's your final answer? [12:14 AM] Corey Couillard: 2320-45t [12:14 AM] Kate Sims: Remember you distributed the negative, so now it's -1000 [12:14 AM] Kate Sims: (in other words, we're subtracting off the head start from how far he has left to go) [12:14 AM] Corey Couillard: 320--45t [12:14 AM] Kate Sims: Yes. Good [12:16 AM] Corey Couillard: so if i was to write out all the equations i would just increase t by 1 [12:16 AM] Kate Sims: I'm not sure what you mean [12:17 AM] Kate Sims: What we have is: [12:18 AM] Kate Sims: Tortoise: distance run: y = 1000 + 45t distance to go: f = 320 - 45t [12:18 AM] Kate Sims: Hare: distance run: y = 360t distance to go: f = 1320 - 360t [12:21 AM] Kate Sims: Are you still confused on the equations? [12:22 AM] Corey Couillard: distance run is the d and t from the starting line and distance to go is the distance and time from the finish line [12:23 AM] Kate Sims: distance run we were calling y. time is t. distance to the finish is f. Do they want time to finish? [12:24 AM] Corey Couillard: Equations for the runners relating time and distance from the finish line [12:25 AM] Kate Sims: That's the f = equations (f = distance from finish line, t = time so it relates those 2 things) [12:25 AM] Corey Couillard: equations for time and distance from the starting point is the y equation [12:26 AM] Kate Sims: Exactly [12:26 AM] Corey Couillard: ok.....I am also being asked to find the domain and range for the graph [12:27 AM] Kate Sims: For both equations together, or each one individually? [12:27 AM] Corey Couillard: together [12:28 AM] Kate Sims: Let's start with range, which is the allowable values for y. [12:28 AM] Corey Couillard: i have to graph all equations on the same coordinate plane [12:28 AM] Kate Sims: Okay, I ask because if taken individually, the domain and ranges would be different for each individual equation... [12:29 AM] Kate Sims: So looking at the range (all values of y), what's the smallest y on your graph, and what's the largest y? [12:29 AM] Corey Couillard: 0 [12:30 AM] Kate Sims: is the smallest. What's the largest? [12:30 AM] Corey Couillard: 1320 [12:30 AM] Kate Sims: I'm going to write the range in the correct from on the whiteboard. [12:31 AM] Kate Sims: Then we need to do the domain, which is the x (or in this case, t) [12:31 AM] Kate Sims: The easy question is what is the lowest number for x? [12:32 AM] Corey Couillard: 0 [12:32 AM] Kate Sims: Now for the highest, it's a little trickier. Where would that be on the graph? [12:32 AM] Corey Couillard: 7 [12:33 AM] Kate Sims: How did you choose 7? [12:33 AM] Corey Couillard: 8 [12:33 AM] Kate Sims: Why? [12:34 AM] Corey Couillard: 7.xxmins [12:34 AM] Kate Sims: If they want an exact answer, then you'll have to figure out how far you really needed to make your x-axis go. What does the point the farthest to the right represent? [12:35 AM] Corey Couillard: minutes it would take for the tortoise to get to the finishline [12:35 AM] Kate Sims: Good. Now we just need to find that exact number. Any ideas? [12:36 AM] Kate Sims: Hint: If f is the distance to the finish line, and f = 320 - 45t, what would I put in for f to represent that he's at the finish line? [12:38 AM] Corey Couillard: 1320 [12:39 AM] Kate Sims: That would be true if we were plugging in for y (distance traveled). You could certainly plug in y = 1320 into y = 1000 + 45t. But how much distance is left when he finishes? [12:39 AM] Corey Couillard: 0 [12:39 AM] Kate Sims: Good, plug that in for f and solve for t [12:39 AM] Kate Sims: 0 = 320 - 45t [12:40 AM] Corey Couillard: 7.11111 [12:40 AM] Kate Sims: Or exactly, 320/45, which reduces to... [12:41 AM] Corey Couillard: 64/9 [12:41 AM] Kate Sims: Yes, good. So I have the domain written in the correct format on the whiteboard [12:43 AM] Corey Couillard: should that be 71/9 min [12:43 AM] Corey Couillard: or should i leave at 64/9 mins [12:43 AM] Kate Sims: Either 64/9 or 7 and 1/9. I just wrote it as an improper fraction (which is how you originally gave it to me) [12:44 AM] Kate Sims: It just depends on whether your teacher has a preference [12:47 AM] Corey Couillard: last question is what will be the margin of victory, both time and distance [12:48 AM] Kate Sims: So we figured out it takes the tortoise 7 1/9 minutes. For the margin of victory, we need to figure out where the hare is after 7 1/9 minutes. How far is he from the finish? [12:49 AM] Corey Couillard: hare is 1320, tort is 320ft away [12:50 AM] Kate Sims: Wait a second, maybe I'm getting tired, but something doesn't make sense. Can you tell me how you got the 320? [12:51 AM] Corey Couillard: tortoise started the race at 1000ft, so he only had to go 320ft versus the hare started 1320ft away [12:52 AM] Kate Sims: Yes, but that doesn't tell you about how much of the difference they made up. [12:52 AM] Kate Sims: We found how long it takes the tortoise to finish. Find how long it takes the hare to finish. [12:54 AM] Corey Couillard: 3.66666 [12:55 AM] Kate Sims: Or exactly, 11/6 or 3 2/3. [12:55 AM] Kate Sims: After 11/6 min, how far from the finish line will the tortoise be? [12:56 AM] Corey Couillard: how are you getting 3 2/3 isn't it 1 5/6 [12:56 AM] Kate Sims: What did you divide to get the 3.666666? [12:57 AM] Kate Sims: Sorry, I meant 11/3 or 3 2/3 [12:57 AM] Corey Couillard: 1320/360 [12:57 AM] Kate Sims: So reduce 1320/360 [12:59 AM] Corey Couillard: I am fine with the 11/3 [12:59 AM] Kate Sims: So now use it to figure out how far the tortoise is from the finish line at t = 11/3 [1:03 AM] Corey Couillard: 1165ft [1:03 AM] Kate Sims: He's gone 1165 ft. which means how far does he have to go? [1:04 AM] Corey Couillard: 155 [1:04 AM] Kate Sims: Good. So the hare wins by 155 feet. To get the amount of time he wins by, take the tortoise's time to finish and subtract the hare's time to finish. [1:06 AM] Corey Couillard: 3.45 [1:07 AM] Kate Sims: I'm a stickler for exact, I don't know if your teacher is. Why not get a common denom and subtract exact? [1:09 AM] Kate Sims: It should be 31/9 [1:09 AM] Corey Couillard: 31/9 [1:09 AM] Kate Sims: Good. Is that it? [1:11 AM] Corey Couillard: time margin of victory is 3 1/9 and the distance margin of victory is 155ft [1:11 AM] Kate Sims: Yes, that's right [1:11 AM] Kate Sims: Poor tortoise, that huge head start didn't help that much [1:12 AM] Corey Couillard: that is it....will you be on-line if I go thru notes and have anymore questions [1:12 AM] Kate Sims: It's 1am here, so unfortunately I need to go to bed. If you have more questions, you can send me a message and we can meet online tomorrow. [1:13 AM] Corey Couillard: ok thank you, good night [1:13 AM] Kate Sims: You're very welcome!