Based on sales data over the past year, the owner of a DVD sto...

Question

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

For what prices is the demand elastic? Inelastic?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A fruit vendor has established that the demand for apples can be described by the function AFP is equal to 30 minus 1.5%, where AFP is the number of apples sold per day at a price of dollars per apple. Identify the price ranges where the demand is elastic and inelastic. Awesome. So it appears for this particular problem, we're asked to solve for two separate answers. We're trying to ultimately figure out the price ranges where the demand is elastic. That's our first answer, and inelastic, which is our second answer. So ultimately, once again, we're trying to figure out what the price ranges where the demand for these particular apples are elastic and inelastic in nature. Awesome. So with that said, our first step in order to solve this problem, first off, we need to define, we need to define all of our known variables and the target variable that we're trying to focus on solving for. So we're told that for this particular problem that our known variables are given to us as the demand function, and the demand function once again states that A of P is equal to 30 minus 1.5%, fantastic. And our target variable that we're trying to solve for is we're once again we're trying to figure out the price ranges where the demand is elastic, which I'll you know put quotation marks around E, and we're also trying to solve for inelastic, which I'll call IE for short, for inelastic. So those are our two variables that we're trying to solve for is our elastic and inelastic. Values for when the price ranges are elastic and inelastic for the demand for these apples. So, with that said, we now need to determine let's focus our attention on solving for the elasticity of demand. So we need to solve for the elastic case. So when the demand is elastic, that's what we're focusing on. So the elasticity of demand, we'll call this equation. EFP. And EFP will be, as we should recall, a note will be written or could be found using the following formula, which states that EFP is equal to DA by DP multiplied by P divided by A, fantastic, and once again, we know that A denotes our apples. Awesome. So now we need to find the derivative of A of P with respect to P, that's our next step. So when we do that, we will find that DA. By DP is equal to D by DP. Of 30 minus 1.5%, which when we take that derivative, we will find that it's equal to -1.5. Fantastic. So now we need to substitute in our known variables, or known values into our elasticity, or elastic formula, and then we need to simplify. So when we do that, we could state that E of P. is equal to -1.5, multiplied by P divided by 30 minus 1.5%. Fantastic. So now, We could go ahead and take it a step further, and we could write that E of P is equal to -1.5 multiplied by P divided by -1.5, multiplied by -20. Plus P. Fantastic. So when we simplify, we will find that EFP. is equal to P divided by P minus 20. Fantastic. So now our next step is we need to analyze the feasible price range. So going back to consider our initial demand function that we're given, which is AFP. is equal to 30 minus 1.5%. So focusing our attention on the demand function, we need to note that for the demand to be a non-negative number, so in order for it to be a non-negative number, A of P must be greater than or equal to 0. So that will mean that we need to take 30 minus 1.5% is greater than or equal to 0. So when we rearrange this equation to isolate and solve for P all by itself, we will find that P is less than or equal to 20. And once again, we're just solving for our inequality equation here, so we're rearranging and isolating this expression to solve for P. And when we do that, we will find that P is less than or equal to 20. And we need to note that at, let's make a note of this, that at P equals 20, we need to note that at P equals 20, the elasticity function will be undefined, so put So, undefined, so we'll put U for undefined. So therefore, the feasible price range will be 0 is less than P and P is less than 20. So the demand will be inelastic, and let's write this in blue. So the demand will be inelastic when -1 is less than E P and EP is less than 0, and this will mean that the percentage change in the quantity demand is less than the percentage of the change in price, as we should recall. So, as we should recall a note that in order to find the range corresponding to the inelastic demand for Once again, for the condition where 0 is less than P and P is less than 20, we need to solve for the following expression, which states that -1. is less than P divided by P minus 20, which is less than 0. So when we multiply all the parts of the inequality by P minus 20, where P minus 20, and let's make a note of this off to the side and In red here because it's very important to note, we need to note that P minus 20 cannot equal 0. And we need to note that P minus 20, and let's put this in parentheses, at P minus 20 is less than 0. And since P minus 20 is less than 0, the direction of the inequalities will reverse as a result. So, with that said, we can now solve for our inequality expression. So when we do that, when we start to rearrange here, Or rather, not rearrange, rather solve, we will find that negative P minus 20 is greater than P and P is greater than 0, and 0 will be less than P and P will be less than 20 minus P. And 0 will be less than 2% and 2P will be less than 20. And ultimately, 0 will be so 0 will be less than P and P will be less than 10. OK. So, therefore, the demand is inelastic for the prices in the interval. So let's make a note of this. So it's inelastic, which we denoted as I, so it's going to be inelastic for the interval 0.10 inside a parenthesis. Awesome, which parentheses indicate open, whereas a square bracket indicates closed. So once again, the demand will be inelastic for prices within the interval 0.10. So, With that said, and let's put a box around that because that's one of our answers that we're trying to solve for. Hooray, we did it. We are on a roll here. So now our next step is we need to recall that the demand will be elastic when E of P is less than -1. So this will mean, as we should recall a note, that the percentage change in the quantity demand will exceed the percentage change in price. So, and we need to now at this point we need to recall a note that we need to find. The range corresponding to the elastic demand for 0 is less than P and P is less than 20, so meaning we need to find P divided by P minus 20. is less than -1, so we need to multiply all parts of the inequality by P minus 20, where once again P minus 20 cannot equal 0, and P minus 20 is less than 0. Due to the fact that since P minus 20 is less than 0, the direction of the inequalities will also reverse as a result. So you go ahead and write that P is greater than negative minus 20, and P will be greater than 20 minus P. and I'm gonna skip a few steps since you know the gist of what we're trying to do. We're trying to isolate and solve for P and when we do, we will find that P. is greater than 10. And that is when you isolate and solve this inequality solving for Pete. So therefore, as a result, the intersection of P is greater than 10, and the feasible price range, and once again, let's note that the feasible price range that we're focusing on is 0. is less than P and P is less than 20. So once again, the intersection of P is greater than 10 in the feasible price range of 0 is less than P and P is less than 20. This will mean that the intersection will be 10 is less than P and P is less than 20. So therefore, the demand will be elastic, and once again, let's note that we're using the letter E. Or elastic. So therefore, the demand is elastic for the prices within the interval parentheses 10.20. And that is it. Those are our final answers. Hooray, we did it. So therefore, without a doubt, Our final answer, let's make a quick summary of everything that we've learned. So, in the end, our final answer without a doubt, will be The demand for so the demand will be elastic for the prices in the interval, 10.20, while the demand will be inelastic for the prices in the interval 0.10. And that is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.
For what prices is the demand elastic? Inelastic?

Key Concepts

Demand Elasticity

Calculating Elasticity

Critical Points in Demand Function

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