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3.5 Continuing with demand function of price for selling umbrellas given by f(p) 3000 100p where f (p) is the number of umbrellas that can be sold at a price of p d...

Question

3.5 Continuing with demand function of price for selling umbrellas given by f(p) 3000 100p where f (p) is the number of umbrellas that can be sold at a price of p dollars per ubrella. There is related function Galle the Elastieity of' demand. E(p) which measures how demand changes with respect to price. The domain for E(p) cannot be bigger than that of f(p): What do you think the domain of E(p) is?

3.5 Continuing with demand function of price for selling umbrellas given by f(p) 3000 100p where f (p) is the number of umbrellas that can be sold at a price of p dollars per ubrella. There is related function Galle the Elastieity of' demand. E(p) which measures how demand changes with respect to price. The domain for E(p) cannot be bigger than that of f(p): What do you think the domain of E(p) is?



Answers

The demand $x$ and the price $p$ (in dollars) for a certain product are related by $$x=f(p)=3,000-30 p \quad 0 \leq p \leq 100$$ Express the revenue as a function of $x$.

Be equal to 25 times eight of the power negative X upon 20. Where X greater than and equal to see you X less than and equal to 20. So here in description we have to use given equation which express price P as a function of demand acts to find the function of F F P. That express demand acts as a function of price P. So we have to give the domain of F F. P. So for finding F F F F P. First of all, we have to express the demand acts as a function of price P. By solving the question for X. So here it is, the power negative Acts upon 20 equal to be upon 25. You can write like that. Now we have to apply logarithmic property and we get Negative acts upon 20 equal to natural Opie upon 25. Now here again we apply lot of ethnic property which is natural O. M upon. And so we get xnegative acts upon 20 equal to Natural Opie negative Natural look 25. So here we have to multiply both sides by -20 so we get X equal to 20 times Natural Law 25 Negative Natural Opie. So we can say X equals do F F P equal to 20 times Natural Law 25 Negative Natural Opie. And here we know that the domain of F O. P is 25 upon E less than and equal to B. And here be less than an equal to 25. So now we simplify this and we get B greater than equal to 25 upon E approximate value 9.2. No here p less than an equal to 25. So it is our final answer.

For the given problem We can consider this demand function. We're now it's going to be 10,000 And then we'll have 1 -3 over three plus E to the negative 0.1 X. The slight changes from the previous problem, we want to find the numbers of units sold for prices when p equals 51,500. So we could do this graphically. Another option would be for us to solve this. I was your basically. So if we put this as 500 We can then divide this by 10,000. And then what we would do is we would add one or subtract one that is multiplied by a negative ones that becomes positive, this becomes negative And then we can divide by three. We would flip the fraction subtract three and use the natural log to get our final result.

So we're given this equation where P. Is priced and exes quantity. So if we want in six and a. Mm. A model that expresses the revenue as a function of P role articles. Well how remaining we have multiplied by that price? So RFP if we want in terms of price we replace the X. With what it is here. And so we multiply p. Times that when we get negative peace word plus 500 P. Now in part B. If we want to find a domain and we assume that our is not negative. Well let's go ahead and use this house and autograph this negative 20 X. Squared plus 500 X. Um And so I had already stopped my bound so that I would fit nicely. Um You may need to scroll and scroll out in order for it to actually find this graph when you do this. So what's the domain? Well if it's not negative that goes from zero up to 25 mm. And the price that will maximize is that middle part 12 50. How much are you gonna make? You're gonna make $3125 now. In part E if you're going to find how many units you sell at this price. Well we take 3125 and divide by 12.5 and I got 250 units need to be sold at this price. Her f since graphic. So if I'm going to graph this I need some key points. So the key point here is 00 And this other key point over here is 25 comma zero Morris. This point here 12.5 comma 3125 It's gonna go up. Uh huh. And the chief says well if I want to earn at least $3000 then words are 3000 marks. Well it's right here and right there. So we need to charge change between $10.15 dollars in order to get $3000. So that point is right here and right there. 15 comma 3000. Mhm. Which would mean that you do need to sell between 203 100 units. That's not one of the questions that's kind of extra. There you go.

So to find the funk. This function as a function of P. Or of P. We basically have to isolate the X. And put all this on this side. So that way peas in your equation. So the way we do that, you have to remember one key thing. The way you get rid of a natural log is with E. To the L. N. X. We'll isolate your ex. So here we want to raise both sides to the power or to the power of P. So the way we do that is you to the P equals E. To the Ellen 500 minus five X. So now we have this right this equation right here. So eat to the Ellen of something. We'll just leave you with what's in your exponents. So that equals 500 -5 x. So now you would sell, you would do what you usually do to attract 500 by both from both sides. And that will give us E. P -500 equals five X. And then you divide by five. These two cancel each other out and you have E. P. Or negative five And you have EP -500 Over a -5. That simplifies to 500 minus E. To the p over five equals X. Or your function with respect to pee. So the way you would find the domain, you have your domain in the equation. B. You have your domain in the function from zero to 90. And what you do basically what we found was the inverse of this equation. So what you do is you actually reverse it. So the way you would find the new domain is you take the original range. So you plug in zero for X. And this original function and the 90 for X. And this original function. And that will give your range or your Y values between the value X. Values of zero and 90 for the original function. And then for the inverse function, you just reverse it so you're new Domain will be the range of this function, which if you plug in those values, you get 3.91. Let's do it 3.91 Greater than equal to X. running out for greater than equal to 6.21.


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