Solve each problem. See Example 2. Two planes leave Los Angeles a...

Question

Solve each problem. See Example 2. Two planes leave Los Angeles at the same time. One heads south to San Diego, while the other heads north to San Francisco. The San Diego plane flies 50 mph slower than the San Francisco plane. In 1/2 hr, the planes are 275 mi apart. What are their speeds?

Accepted Answer

Hey, everyone in this video, we're gonna be working through the following problem. Two cars A and B left at the same time but towards opposite directions. Car A moved towards the east and car B moved towards the west. The speed of car A is 20 MPH slower than the speed of car B, After one hour, both the cars are 120 miles apart. What is the speed of car A and car B? We're given four answer choices. Option A car A is 50 MPH and car B is 70 MPH. Option B car a is 40 mph and car b is 60 mph. Option C car A is 70 MPH and car B is 50 MPH and option D car A is MPH and car B is 40 MPH. Now, we're gonna start by drawing a little diagram of what's going on. Ok. So we have our two cars, I'm gonna draw them as boxes. So we have car A and we have Car B and they start in the same position. Ok? But they're gonna move to opposite directions. So car A is moving east to the right B is moving to the west or the left. OK. The opposite direction. Now we're gonna say that the distance car A travels is going to be D A in the distance car B travels is going to be DB. Now, we know that this total distance separating the two of them, the distance between B and a in the far left and the far right is miles after one hour. Now, from our diagram, we can see that D A and DB Make up this 120 miles. So we have the, the distance that car A travels D A plus the distance car B travels is equal to miles. Ok? When T OK, we're gonna say T A is a car A time, TB is a time of car B and both of these are equal to one hour. Ok? We're talking about one particular time point when they're 100 and 20 miles apart. So the time for each of them is gonna be one hour. So the total distance between them we figured out is the sum of the two distances they've traveled since they're going in opposite directions. And we know that that's equal to 120 miles at time. One hour. Hey, now we're looking for the speed of car A and car B. So let's go ahead and define our variables. OK? This is really important. It helps people be able to read your solution follow through your work. OK? It can be really helpful if you get stuck or you need to come back to the problem to be able to read what you've done. And when you get to the end of the problem and you get an answer, you can double check with the variables you've laid out and make sure that's what you were looking for. Ok. So we're gonna let V B be the speed of Kirby and we're gonna let va see this speed. Ok? No, we've been told that car a is 20 mph slower then car B. Ok. So VA is going to be equal to V B minus 20 MPH. Ok? The speed is 20 MPH less. Now, based on this information alone, we can already eliminate two of the options. Ok? We know that car A is slower. So the speed should be lower. Option C and option D both have a higher speed for car A than car B. So we can already eliminate option C and D from our answer choices and we're left with just A or B. We're gonna work out the rest of this problem to sort that out. Now, we have some information about the distance, we have information about the time we're looking for speed. So let's recall the equation that relates the three of those together. OK. We have that the speed is equal to the distance divided by the time. And we're gonna write this for each car. So we have that the speed va of car A is equal to the distance D A divided by the time T A. And similarly, we have the speed V B is equal to the distance DB divided by the time TV. Now we have our equation D A plus DB is equal to 100 and miles. We're gonna call that equation one. What we're gonna do here is we're gonna take the equations. We have V is equal to D over T, we're gonna isolate for those D values and then we can substitute them into this equation. OK? We know the times and we're trying to find the speed. All right. So we can write that the distance D A is equal to the speed VA multiplied by the time T A and similarly, the distance DB is equal to the speed V V multiplied by the time TV. And we're gonna call Those two equations, equations 2, 3. Now we're gonna sub those into that first equation. OK. So we're gonna set two and 3 into equation one. What's that gonna give us? Well, it's gonna give us va multiplied by T A plus V B multiplied by TB Is equal to 120 miles. Now, we're looking for these speeds. VA and V B, we know the time T A and TV, we can substitute that in if we do that, we're left with both speeds in this equation. Though. We have VA and we have V B, we can't solve for two unknowns with one equation. Remember though we have information about the speed of A and how it relates to the speed of B. OK. So we can write VA is equal to VB -20 mph. This is multiplied by the time, which is one hour Plus VB multiplied by the time of one hour again. And all of this is equal to 120 months. Ok. So that relationship between the speed of car A and B has allowed us to write this as an equation only of the speed of car B. And now we have one unknown and we can solve, we're gonna start by expanding on the right hand side. We have VB -20 mph multiplied by one hour. And actually let's go ahead and divide by this per hour first. Ok. We have that in both terms on the left hand side. So let's go ahead and get rid of that right away. We're gonna have BB -20 mph. What baby Is equal to miles divided by one hour gives us 120 mph. Ok. Now, let's simplify. On the left hand side, we have V B plus V B That gives us two VB and then we still have our -20 mph. This is equal to 120 mph. Ok. We wanna isolate for V B, we can add 20 MPH to both sides. We get two V B is equal to 140 MPH. And dividing by two gives us a speed of car b of 70 mph. OK. So we found the speed of carbi. Now we know that that's 70 mph. Hey, we need to find the speed of car A as well. Ok. But remember car A is 20 MPH slower. So it's equal to the speed of car B minus 20 MPH, which is equal to 70 MPH minus 20 MPH, Which gives us a speed of car a of mph. Ok. So now we've gotten both of the speeds. We know that car A is traveling 50 MPH. Car B is traveling 70 MPH. If we compare this to our answer choices, we see that this corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Solve each problem. See Example 2. Two planes leave Los Angeles at the same time. One heads south to San Diego, while the other heads north to San Francisco. The San Diego plane flies 50 mph slower than the San Francisco plane. In 1/2 hr, the planes are 275 mi apart. What are their speeds?

Key Concepts

Relative Speed

Distance Formula

System of Equations

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