Problem

In the following exercises, solve the given maxim…

Problem 5 Easy Difficulty

In the following exercises, solve the given maximum and minimum problems.
The power output $P$ (in $\mathrm{W}$ ) of a certain $12-\mathrm{V}$ battery is given by $P=12 I-5.0 I^{2},$ where $I$ is the current (in A). Find the current for which the power is a maximum.

Answer

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Calculus 1 / AB

Basic Technical Mathematics with Calculus

Chapter 24

Applications of the Derivative

Section 7

Applied Maximum and Minimum Problems

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Video Transcript

So for the fall spot, we have to find the art on Dhe. Let's first write down the information that we already have. So we have power on DDE. We have the e m f all this so on. Let's try to connect these three questions so we know that Powers is equal dough spare off the world stage over Vicky Valent resistance. So let's write down the equivalent a system that's hoping that we will get, uh, this expression you cannot Linda Systems in himself are so that you get violent. Resistance will be seven owns. So this is that a sister, that is in cities, they battle combination off resistors, 12 rooms, four rooms on our. So this that I'm writing. It's just the Vicky Valent resistance off the battle combination off these three resistors. Now we can substitute this expression over here. Ah, like I would suggest you not to worry about this room. Sign on just removed us and solid terrorists off the question because this is the sinus really bother. This time really bothers me while solving the secretion, so yeah, don't worry about this unit. You can love coordinators. So now you can substitute these two values over here so are violent that the equal Joe scare off the M F over power on this will be equal. Do this whole expression. No substituting all the values along with the voltage and power and solving for art. You should get out of the window. 19.5 homes. Now, this is question is really easy to solve. It will. Really Mary question not 1/4 to go on or the polynomial equation. So, yes, this'll question is going to be real easy. You just have to cross multiply all the tones and salt for art. So now for the second Bach, you have to find our So we know that tongue. This a question. We see that followers, What is proportional? Two are violent. So here I am, using power to be equal to scare off the current dying sick of island resistance. So why from this equation you have power to be proportional toe equivalent resistance on dhe. Therefore, in order to minimize, why would Okay, so, uh, okay. Actually, they're using the same question. So this is the question back. We're gonna follow again, supporting the execution. Our office constant. It means that power is inversely proportional to the UK violent resistance, which means that the maximum power will occur are the minimum resistance. So looking destroyed this So be Max well occurred at the minimum IQ Valent Assistance on dhe for the minimum Vicky Valent resistance. And let's take a look at this equation. Now This is possible when all is it called a zero. That that way, the minimum, the equivalent resistance will just be seven wounds. So for that we need our to the equal to see you for maximum power. Andi, obviously minimal make violent assistance. For the third part, we have to find a minimum power, which means that we must maximize the equivalent resistance. Oh, simply began said are to the infinity and that we will get the maximum Vicky Valent assistance and hence the many one powers not for a party since the minimum IQ a violent resistance, as we saw here, the lockers when our physical zero on there would act. Give some stuff gives us that given and resistance to be seven homes so we can substitute this in this equation and salt for power. It's a power to be goingto square off the year. Math, which is 24 would over. But assistance. This gives stall maximum power to the equal door. 82 0.3 words. Then we have to find the minimum power. So for that, as we discussed in part, C. R. Needs to the infinity now. One thing too. Um, notice or take care off is when all was equal to zero. We simply substituted articles to zero over here on salt for our equivalent. But when our physical to infinity, you can't do that because we'll have Ah, he'll have an ended on the note over here. So instead of that, what you can do is begin use logical reasoning, which means that okay, if our is called infinite, it and we have infinite resistance in that wire or at that part on, that means that no current flows two that wire's on that part off the circuit and therefore the only I got it. The only parallel combination. The only resistance is that are in parral combination. Are these two, so we can just eliminate, are here and just find the equivalent resistance off this panel combination on dhe, then find the totality violent resistance so I'm just gonna find that. And this is the maximum again valiant resistance. So that should be called to seven ones That doesn't see these with, well, go on four room resistance. And since no current flows through this part the infinite resistance, so we're not gonna worry about Oh, that And this comes out to be equal to 10 homes. Now that we have this, we can find the many among followers by the same matter. So we divide the square off. Mm, by the maximum of a violent resistance. And we get the minimum tough hours to be 57.6. Thank you.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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