Solve each problem. See Example 2. In the Apple Hill Fun Run, Mar...

Question

Solve each problem. See Example 2. In the Apple Hill Fun Run, Mary runs at 7 mph, Janet at 5 mph. If they start at the same time, how long will it be before they are 1.5 mi apart?

Accepted Answer

Hey, everyone in this problem, Alan and Mark participate in a school race, Allen and Mark Run at a speed of six mph and four mph respectively. We're asked to find the time When both of them are 2.5 miles apart. If they begin at the same moment, we're given four answer choices. Option A one hour, Option B 2.5 hours, option C 1.25 hours and option D five hours. So let's start with writing out what we know, we know that the speed you're gonna say called V of Allen va Is equal to six mph. OK. And the speed the M of mark is equal to four mph. We wanna define the rest of our variables. So we're gonna say that D A is the distance. Do Allen, how's it run? And D M similarly is the distance that Mark let's run? Ok. Ok. So we have our two distances D A and D M and we want to find the time When the distance is 1/2 miles apart. So we want to find tea when the absolute value of D A minus D M is equal to 2.5 miles. All right. So this absolute value, it just means we don't care which one is in front. We don't care which one is bigger. What we care about is that they're 2.5 miles apart. Ok. So we want the magnitude of that difference to be 2.5 miles. Now, what equations do we have that you use these variables? We have speed distance time. OK. So recall we have the equation V, the speed is equal to the distance divided by the time. So we can write this out for each of our participants, we have VA is equal to D A divided by T A. OK. So for Allen, his speed is equal to the distance divided by the time. And we can rearrange this to write an equation for the distance, which is what we're interested in. So the distance Allen runs is going to be equal to his speed va multiplied by the time it takes T A. Now we're gonna do the exact same thing remark. So for mark his speed, V M is equal to the distance D M divided by the time T M we can rearrange. So his distance D M is equal to V M multiplied by T not T ATM. All right. Now, we want the absolute value of these, OK. The difference of these to be equal to 2.5. Now, Alan is running faster. OK. So he's gonna move a greater distance in the same amount of time. So D A is going to be larger than D M. So we can actually drop those absolute value signs. OK. So we're gonna do that. Um Just for simple, simplifying. Um But D A is going to be bigger because Allen is moving faster over the same amount of time. So we have D A minus D N. This is gonna be equal to va multiplied by T A minus V M multiplied by T. And we want this equal to 2.5 miles. Now, we want to find a time when they're 2.5 miles apart. That means the time for Allen and the time for Mark is gonna be the same. Ok? We're looking at one particular time point, so T A is going to be equal to T M and we're gonna call both of them just T. Ok. That's the time that we're looking at where they're 2.5 miles apart. So now we can factor that time T from our equation, we get T Multiplied by VA -3M Is equal to 2.5 miles. VA and V M. We know we can substitute in those values. So we have a T Multiplied by six mph -4 mph is equal to 2.5 miles. We want a soft for tea so we can divide we have t, is equal to 2.5 miles divided by two MPH. Ok, six MPH minus four MPH gives us two MPH. Now, we have miles divided by MPH. That's gonna leave us with a unit of hour. Ok, 2.5 divided by two. And so we get 1.25 hours. Ok. And that's the amount of time or the time when the two of them are gonna be 2.5 miles apart, they begin at the same time. That's the time we were looking for if we compare this with our answer choices. Ok. We found that the time Where they will be 2.5 miles apart is 1.25 hours which corresponds with answer choice. C Thanks everyone for watching. I hope this video helped see you in the next one.

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Solve each problem. See Example 2. In the Apple Hill Fun Run, Mary runs at 7 mph, Janet at 5 mph. If they start at the same time, how long will it be before they are 1.5 mi apart?