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The demand function for certain product is determined by the fact that the product of the price and the quantity demanded equals 5000. The product currently sells f...

Question

The demand function for certain product is determined by the fact that the product of the price and the quantity demanded equals 5000. The product currently sells for $2.70 per unit Suppose manufacturing costs are increasing over time at rate of 13% and the company plans to increase the price at this rate as well. Find the rate of change of demand over time_Find the demand function represented in this problem where is the price of the product and is the quantity demanded for the product(Type an

The demand function for certain product is determined by the fact that the product of the price and the quantity demanded equals 5000. The product currently sells for $2.70 per unit Suppose manufacturing costs are increasing over time at rate of 13% and the company plans to increase the price at this rate as well. Find the rate of change of demand over time_ Find the demand function represented in this problem where is the price of the product and is the quantity demanded for the product (Type an equation )



Answers

The demand function for a certain product is determined by the fact that the product of the price and the quantity demanded equals 8000 . The product currently sells for $\$ 3.50$ per unit. Suppose manufacturing costs are increasing over time at a rate of 15$\%$ and the company plans to increase the price $p$ at this rate as well. Find the rate of change of demand over time.

This is another example of a more worthy related rates problem. So that's first translate the question into simple, like math, sold the question and find the answer. So the man function for this product is determined by the fact product of price and quantity equals 8000. So that's called price speed and quality. Cute. So, um, then this relation is at the product off P and Q equals 8000. Now we know that the product currently sells for 3 50 so the prize is 3 50 And we know that the manufacturing coasts are increasing over time and the rate of 15%. And the company plans to increase the price at this rate as well. Okay, so we're giving given dp DT and slightly roundabout way rather than just telling us what DPD deice. They tell us that the rate of change for the price is 15% over the price or other words 01 15 times. Pete, that is really what what it means. Okay, so now we have almost everything we need to get this into standard related great relief related rates. Question. The only thing we need to figure out is what is. Q. When P is 3.5. Well, their products is 8000. So, Hugh, it's 8000 divided by 3.5. So now we have everything extracted from the question, and we can to our standard differentiate substitute solve relate greats. Algroup so differentiates, UH P Times q equals 1000. This is a product rule we gets. Dp DT multiplied by Q plus P. Times Q B t equals zero s t derivative of a constant issue are okay Now we, uh, get substitutes no substitute or are known values. So the P V t is there a place? 15 p That's B times cute. Um, plus 3.5 q t t. And this equals zero. So now you see, we didn't actually need to figure out que specifically because the only thing we need to know is the product of P and Q. But that is always 1000. So we actually get 0 18 times 1000 plus three 0.5. Thank you needy equal zero. So sometimes these things will simplify your life, so there's actually smarter to just do the differentiation and the substitution as far as you can go before like solving the origin equation to find. Q. So in this case, it wasn't hard and took us like a second. Um, but sometimes it will be hard to do soul and it'll turn out. You don't actually need the specific solution for Q. Maybe only a product of P and Q ass. In this case, we might be easier to sold for. So, uh, we are interested in d. Q. T T. So you saw this guy for the Q t t. And we find that Q T is negative. 8000 times syrup 15 divided by 3.5 and we throw that into a calculator. So if you plug this into your calculator, you find that this is approximately negative. 343. Um, so this is not the answer. That's in the answer section. That is minus 98 which, coincidentally, is about 13 point fifth off this, um, so I think in this case, the answer in the book is actually wrong. I do believe this is the correct answer. Um, what I believe the book for got to do is include this factor p over here. So they said the rate of change is just 0.15 rather than 15% off the price. Um, and if you do that, you actually get the answer in the boat off minus 98. So I do believe this answer is correct. It should be minus 300. Other minus 343. 34. Sorry. Um, so the answer would be that the demand is dropping by 334 units per time.

Question of 65 were given the marginal demand function on. We need to find the total demand function or the demand function. So for to find the X will integrate both sides. So that's how it looks like. Uh, this is X square over here. We got we got to integrate this now in order to integrate 4000 of the constants that comes out Said, using the constant rule Andi, If X square goes on top, the power will become minus two. So using the chain rule, this will be X rays to the poor, minus one over minus one, plus the constant of integration of C Uh, this is D X. So this finally consult us 4000 over X plus C were given another piece of information that when excess four than D is 1003. So let's substitute excess four. It fixes for than 4000 before 1000 and D s 1003 plus c. So from here, see, comes out as three, which we substitute over here. So the final demand equation comes out as 4000 over X, which is 4000 war X plus C where C is three. So this is the final answer

Are they giving up? Um We have that the demand is inversely proportional to the square of the price. P. So when the prices 10, the demand is 2000 500. So we want to find the revenue as a function of X and approximate the change in revenue. Well, if we know that the demand acts is inversely proportional to the square of the price. P. We know that this is going to be the X equals one over hurt. I want something like this. And we know when the demand is 10. So when when the price is 10. So this right here is price being 10? That gives us 100. 1 over 100. We know X. The demand is 2500. So that means this needs to be um 25. That way when p equals if we have 25 over 100, we want this to now be 25,000 Or 250,000. All right, there would be our model for the given equation.

For the given problem, we want to look at the demand for commodities. So um we have P. Equals square root of a negative X. Square. Mhm. And we know P. Is the price of dollars. So let's suppose the quantity demanded is 6000, so X. Equals six when the unit price is eight. Um So we want to determine the demand equation based on that. Um So we can plug in our value of six and it's going to be the same thing. Medina prices eight. So we're gonna put eight here and we would put A six here. And with that we'd be able to end up finding our values of A. And B. So we end up getting this is going to be a negative X. Squared right clocks 100. And that would be our demand function.


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