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Find three numbers in AP whose sum is 21 and product is 315 ....

Question

Find three numbers in AP whose sum is 21 and product is 315 .

Find three numbers in AP whose sum is 21 and product is 315 .



Answers

Find three positive numbers whose sum is 3 and whose product is
a maximum.

Talk about this question, we have to find the sum of all the teachers between 21 and 45. So the first even interior is 22, then we have 24, then we have 26th and we have 28. And this will go like this still the last even if it's just gonna be 44 because 45 years old. So this is nothing but an arithmetic progression, arithmetic progression with the first term as 22. The common differences 20 for -22, which is two. And uh the number of terms are unknown and the sum is also unknown. So to find a number of terms. First we got to use uh in the in its term next term is 44. So to find the number of terms we're gonna use the equation, A N is equal to a plus N minus one times deep, which means that that's probably equal to 44 is equal to 22 plus n minus one times two. Uh So this will be equal to 22 is equal to two times and minus one, dividing both sides by two. That's going to be 11 is equal to n minus one, Adding one boat, says it means that any school 12. All right, so, and as you go to 12. Now, we can easily find the some of these numbers of some is going to be equal to a number of times over two. And that's first term plus the last number of times are 12, 12 over to the first one is 22 and the last term is 44. That's going to be equal to six times, uh, 66 6 times 66 or 66 times six is 3 96 3 96. So this is the required some of these numbers. Thank you.

We're going to find two positive numbers whose product is 147 and the some off the first number plus three times. The second number is a minimum. Yeah. So in this type of problems, we get to transform these, uh, sentence into equations. For that, we get to use names or variables for the expressions or quantities who want Thio calculate. Then we gotta transform the sentence into equations using those names or variables. Right Then we tried to write the equations into a one single variable problems. Would that be Maybe gotta use some algae, Iraq or manipulations of direct manipulations off the expressions or equations obtained before and after that. We get to use some calculus too. Obtain the minimum value in this case. So we start with given names to the quantities are which are involved in this statement. So we want to find two positive numbers. The Boston number has properties in this case that the product of the numbers is 147 and the some of the first number plus three times the second number is minimal value. So we have two numbers. So the natural way to start is to give names or rival names to these numbers. So Leg X and Y be the two positive numbers we are searching for. Look it doing this. We can now write a sentence in terms of equations. So we have the first properties that the product of the numbers is 147. So x times y equals 147. And this is our first equation. Now the some of the first number plus three times the second number, that is X blood three. Why get to be a minimum or get to be at its minimal value for the X and Y we give a solution So we can say that this expression we're going to call it s got to be a minimum. So we want to minimize this expression. So that's our goal. In this case, we want to minimize t expression s equal eggs plus three times why, and this is fresh in depends on two variables in this case, positive numbers. So in this case, we want to, or we have to imposed the condition that X and Y are positive for ex President zero and why created than zero This is the translation of the fact that the two numbers are positive. So we have two variables and we have thes and the main for the variables positive X and y Right. So to manipulate this problem with the calculus of one variable, we get to transform this equation into an equation off a single variable. And for that, we used the condition off or the condition that say's that the product of numbers is equal to 147. So from x times y equals 147 we can solve for X Y. In this case, we solve for why and we get white was 147 over X. We can do this division because they got a bit different from zero, because if some off x y r zero, then the product cannot be 147. So this'll division here is well defined. And now, given this expression, okay, we get that s is now a function of one variable Onley variable eggs. Because when we plug this value off, why this expression into discretion off s we get s dependent on Lee on X. So we get X plus three times Why? Which is 147 over X. That is our expression that we want to maxim to minimize is given by ex plus three times 140 seventies 400 41 over X four eggs bust. So we want to find Yeah, yeah, yeah, a value of X positive or positive eggs such that. Okay, as of X is minimum and or putting it another way. In other words, we get to find the value of X Were SFX attains its minimum value for positive X. And now for that, we calculate the derivative of s. And this is one okay minus for 141 over X square. This is our derivative. And so the derivative of S zero, that is, we're looking for critical points, Bess, as even on Lee. If one minus 145. 41 story over X square zero me. That is well 441 or x square is one or what is the same. X square is 441 and this'll This implies that X is 21 because eggs is a positive body. This solution another solution of this equation is negative. 21 but we only our interest in positive solution. Yeah, so, having this, we now can say to follow it. The derivative of s can be written, remember, Is discretion here one minus 441 over X square And this is X square minus 1 441 over X square and the denominator X squares. Positive. So the sign off the derivative depends on the side. Sign off the numerator X square minus 441 and haven't written the derivative off. Asked this way, we can see easily that for X positive and less than to anyone. Because for positive value thes the square functions X square is increasing. We can say this right. Remember the squares 21 is 440 41 then this implies is X squared minus 441 is negative. So for X between zero and 21. The numerator of the derivative of s is negative. And because the denominator is positive, we said before the derivative is going to be negative. And so after everybody is negative and poor X greater than 21 for the same reason as before. There is The square function is increasing for the positive. He simply these which implies that X squared minus 441 is positive. And then, uh, the derivative of s is positive. Okay, What we have here is that for the interval zero from 0 to 21 t derivative is sorry. The function is decreasing because it's derivative is negative. So ISS is decreasing. Yeah, on the interval 0 21 and is increasing on 21 infinite infinity because there is relative it's positive for experience than 21. So if the function is decreasing from 0 to 21 increasing from 21 to plus infinity Yeah, we must have that key functions has a minimum at the in fact, an absolute minimum up 21 for the positive X values then as yes, an absolute minimum at X equal 21. When is consider Yeah, for Ex Post, which is our case for 40 is numbers we're looking for. Our must be positive. So 21 in fact, is where the function attains its minimum value for the positive X. Then we have this and now we want to find why. Remember, we want two numbers. The first one we have found already is 20 one. We need the other one, which comes from this equation here where we wrote. Why, in terms of X, that is why is 147 over X. So now why is 147 over eggs? That is 147 over 21 and this division is equal to seven. That is why is seven. So we have our answer way. Have that. Okay. The two positive numbers, Um that satisfied the Cuban requirements. Hi are 21 and seventh, That is Yeah. For these numbers we have that the product is 147 on the some off ex, uh, off 21 plus three times seven is a minimum body. And so this this dissolution off the given problem here the

Today we're going to solve Called the number 33. Yeah, we have X plus y plus 30 was 30. Is that because well, 13 minus X minus money like that. Sickles, ex wives that with physical X way into 13, minus X minus Y we get like that DX way minus X squared away minus. It's like quit. Yes, with respect to access 30 X minus two Excellent minus y squared. Assert respective ways that the Y minus X squared minus two works well. S X equals zero and it's the way it was zero that the x minus do It's like minus y squared equals zero that the Y minus Take the square minus to exploit equals zero has a solution and go on then and minus 32.75 Coma 2.75 So as double to is 30 minus two way as why why equals 13 minus two weeks as X y equals minus two. It's minus two white so we can see that as sex at 10 come or 10 is greater than zero is x six at minus 32.75 Coma two points and fight greater than zero does not exist. Maximum value does not existed. Maximum valuable. Thank you

We have to find the some off the even in teachers between 21 on 45 So therefore our first time will be 22 on our last time will be equal to 44. We know that the general or the end its term is equal to a one plus and minus one multiplied with D. Since we're finding even in teachers, so therefore, the common difference will be equal to two. Now the end. It's Thomas 44. Let's find end so we'll put a an equal to 44 which will be equal to 22 plus and minus one multiplied with two that is 22 will be equal toe to end minus two, which will give us the one is equal to 24 which means N is equal to 12. Now we know that the summer fen terms often arithmetic sequence has given us in divided by two multiplied with a one plus in to putting the values off n a one and a N will get 12 divided by two multiplied with 22 plus 44 which is equal to 396


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