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Solve each problem. An accountant determines the charge for filing a tax return by using the function $C=50+40[t]$ for $t>0,$ where $C$ is in dollars and $t$ is in hours. Sketch the graph of this function. For what values of $\ell$ is the charge over $5235 ?$

Problem. 95. Um, we're filing, ah, tax return. And the charge for filing that tax return is C equals 50 plus 40 t where C is the charge in dollars and t is the time and ours all right. And we have the greatest integer function in T. Okay, So if that's ah desk, more fraction, it gets rounded down to the nearest integer. And they want to know what values of T is the charge over to 35. Okay, so we want see greater than 2. 35. This is our function right here. Dysfunction right here. See? Okay. So I am gonna substitute that function in place of C. All right. So in place of, say, 50 plus 40 t is greater than 2. 35 minus 50 on both sides. Fifties cancel 40 t is greater than 1 85 and dividing both sides by 40. We have tea is greater than 4.6 to 5. Now, remember, for tea, we have the greatest integer function. Okay, so this decimal is always going to get cut off because it rounds it down to the nearest integer. All right. So because the decimal is always rounded down. We need to round tee up because he has to be greater than this decimal. But if it even if see this is greater than 4.625 let's say it's 4.9. It's still going to get rounded down to four. So that's not gonna work. So we need to round it up. So we get rid of that decimal, and we say T is greater than five. Okay, and that makes perfect sense. If you plug in five for tea, you'll get 40 times five, which is 200 plus 50 is 2 50. Okay, but if T is less than five, it will get rounded down to four. You know that four times 40 is 1 60 plus 50. That's to 10. That's no good. That's not greater than 2 35. So it has to be five or more so, really, I should have said greater than or equal to five. Okay? And they also wanted us to sketch the graph. So I just counted by ones for the X axis that's ours, and I counted by forties for the Y axis. That's cost. It's because it goes up $40 every hour. Okay, so up to one hour T is gonna be zero, and it's just $50 upto one hour. So we start. We start right here upto one hour. Okay, That's gonna be $50 all right? And it goes up by 40 each time. And you know what? I sink. I think it will be better if I rewrite this y axis and start with a 50. So let's do that. I'm gonna start with 50 on the white axis. That will make things easier. All right. So this will be 50 then we'll count by 40 each time. 90 1 30 1 70 to 10 2. 50 and so on. And we'll pretend Will say it started at 10. All right. The X axis started at zero. Okay, let's try this again. And I think I'll make it in red so you could see it better. Okay, so for the first hour, it's $50 for the first hour. All right? And if x 50 equals one all the way up to 1.999 it's gonna be $90. Okay, So I'm gonna make a open, open circle here, and it jumps up to $90 at one all right, and then make an open circle at two, and it jumps upto 1 $30. I make an open circle at three, and it jumps upto 1 $70. Open circle at four. It jumps up 2 to 10 and so on. It keeps going. Keeps going up and up like steps. All right, so that's this. That's the sketch of the graph. Each time it goes up to the next hole, our it jumps up by $40. Okay, but any fraction of an hour gets rounded down because this is the greatest integer function, which means it's rounded down.

First, let's identify which charges or positive and which ones are negative. Positive judges have build lines pointing out. Negative charges have field lines pointing in. So a has broad lines burning out that makes it positive. Be his field lines pointing in That makes it negative. See his building 20 out? That makes it positive. Okay, now we can't the field lines for each. So a has 1234 four field lines going out. B has 12345678 and see his 12345678 because the Nuffield ones can only tell us how comparatively strong the churches are. The actual char charge regard to note with an X so that we can use algebra to solve for it. So a was positive for X vehicles. Negative eight X c was positive. Eight X Now we know that Q. Is this some of these charges, so Q equals a plus B plus c. So we have Q equals positive four x minus Beta X plus eight x. Okay, so you have this all together Q equals four X. And then we're just solving for X ser que over four equals X. So now we just put those back in to a B and C. So we have a equals four X b equals negative. Eight X c equals a X, but X is cured before, so a equals four que over four, which is just cute. B is going to equal negative eight que over four for negative two Que and C is going to be a Q over four. Or tu que you should put these days here. So that's the charges of a P and C. And if you want to check your work, you can add them all together and see if they had to queue, which is what we originally said they would appear. So if q Plus two Q minus two q equals Q. So we know that answer makes sense.

In this problem We're going to be calculated the electrostatic force on Q one. The charge of the origin from the other two charges Q two and Q three positioned as shown in the diagram, we are given all the charges mm Norris in particular. Thank you to and Q three same AG two opposite assigned very important. Going back to the diagram noticed that Q two and Q three are both the same distance from Q. One and also the lines connecting those charges to Q one are making the same angle with the positive X axis. All very important. So let's now start looking at the forces that are on Q one. Q one positive to negative. That means an attractive force. That means the force of two on 1. It's going to be in this direction. Q one positive to three positive. That means that forces repulsive along this line. So this is F 31 for future use. If this angle here is 23°,, then this angle here also will be 23 degrees. No if the charge managers are the same and their distance from the other charge that you're looking at, the force on is the same. And that means command to that force forces will be exactly the same. So let's make that explicit. So let me write the expression for the charge to one. The decay times the magnitude of charge too, magnitude of charge one over are 12 square Now let me ask what's the same. Okay, certainly it's just a constant Q two and Q three mantras are the same. So I can replace Q 292 to 3 million to Q one is still the fourth charge. We're looking for the fourth song let's say the same. Are 12 and R 13 are both 1.3 m. And so this is our 13 squared now. Well, that looks to me to be The magnitude of the force of three on 1. So they are the same. No, we'll need that right now let's look at F net X. Now that's going to mean we need F 21 X and F 31 X. We'll have 21 X. Trying to triangle Yeah, X. component is going to be the adjacent side to 23. So that and it's in the power of direction. So this is going to be F 21 Koh Sai 23°.. I have 31 that's in the negative direction. And and the X component is again the adjacent side was going to be minus F 31 Co side 23°.. But I just mentioned and just showed you the 21 man, 231 man to the same. This is zero. There is no net NX. And I could put an E. On here indicating electrostatic because we know there's mechanical forces but we don't we don't need to do that. So now let's look at F net. Why now in both cases, yeah, they are the opposite side of 23 degree angles and they're both in the positive Y direction. So it's going to be F 21 sign 23 degrees plus F +31 notice. There's gonna be no cancellation here. So it's gonna be too F to one side 23 degrees. So we now have announced your value and we can put in our formula here. So this is gonna be too times the bank to k. Times the magnitude of cute too Overseitude of Q one sign 23 degrees over. See what I called L squared just to make it so that that it's easier to remember that they were the same. And we can put in our numbers two 8.99 times 10 to the nine. No. 10 major square Coolum squared and charge two in magnitude is five Times 10 of them out of six columns. Q one is 8 times 10 to -6 Cool homes and don't forget to sign 23 And that is over. That is over 1.3 m Squared. And this works out to be 0.166 new tints. And seeing that this vector only is in one direction only in the Y direction. This is also this is F net the magnitude of that factor. So that is the magnitude of the net force and Q. One and the direction to make an explicit direction is plus why and that we were already knew once F net x zero. So that's the direction. So that was part A. That was part A. To get the net force a part B. They wanted to know what the acceleration would be. Remember when they see a celebration they talk about a vector means magnitude and direction. Uh huh. So we know once we have the net in this case is only F net. Why? That M A. Y. So a Y. Is going to be F net Y over em. And that will be 0.166 Students. And they give they give a mess of 1.5g, which is 1.5 times 10 to the -3 kilograms. And this works out to be 111 meters per second squared. The man who had been 111. The direction because it's a plus that implies plus why direction? So we have so we have 111 m per second. That's the magnitude in the plus five direction. It's all kind of combines all of you, so that is the acceleration and that's the rest of the problem.

So for this question, um, we can start five. Figure out the diagram for this, sir. Um, for Q one and Q two, there actually were equivalent charges on opposite ends. Equal distance away was four centimeters, four sentinels. So for this problem, we don't actually need to look at those two for when it comes to the Net electric field, since they cancel out then electric Field zero so they don't really effect the second part. Q three and Q four we have Q for at seven centimeters. Q. Three, five centimeters. So to do this one, um, it's except the equation for the two tribes when you're fine. Okay, constant times. Q. Must be to three over the distance squared, and we do that for the second leg. Look. This time it's for Q four. This one will be. This is for three that before, and they're square of the distance is square. So when you just plug in the numbers, you can fact around K, which is nine times in the night. The two charges distance is fat and 72 meters, uh, squared, plus the other charge negative eight times 10 to the negative. Six Coombs we're seven to the distance. So the two charges three times and Mega six the negative 8 10 7 acres, six schools and are the distances on the bottom which are squared. Find 10 to the negative media's. We convert two meters from centimeters. And so, uh, with all these plugged in, you get your value. The magnitude of the electric field is 3.9 times 10 to 6 and this would be in the positive. Why direction?


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