A deli sells 320 sandwiches per day at a price of 4 each....

A deli sells 320 sandwiches per day at a price of $4 each. (A) A market survey shows that for every $0.10 reduction in the price, 20 more sandwiches will be sold. How much should the deli charge in order to maximize revenue? The deli should charge $ for a sandwich to maximize revenue. (Round to the nearest cent as needed.) (B) A different market survey shows that for every $0.20 reduction in the original $4 price, 10 more sandwiches will be sold. Now how much should the deli charge in order to maximize revenue? Now the deli should charge $ for a sandwich to maximize revenue. (Round to the nearest cent as needed.) Enter your answer in each of the answer boxes.

Transcript

00:01 Hi there, so for this problem we are told that delhi sells 480 sandwiches and this at a price of four dollars.

00:13 Now for part a of this problem we are told that a market survey shows that for every 0 .10 reduction then we increase 20 more sandwiches okay.

00:34 So then the question is how much should the delhi charge in order to maximize the revenue.

00:40 Now the first thing that we need to find for this is the demand price expression and for that we use the following.

00:51 So we can consider that the final price for this is going to be a reduction of the price that will be 3 .90 dollars and then we will increase the number of sandwiches to 20 more so we add 20 to 480 so we obtain 500.

01:19 So then we will have the price p minus the initial value or the price that is 14 this divided by x minus the initial number of sandwiches that is 480.

01:31 Then now we evaluate this at the final value so that will give us then minus 0 .1 or 10 then this divided by 500 minus 480 so that will be 20.

01:44 Now we just need to solve for p so p will be then minus 0 .1 divided by 20 then this times x minus 480 and then this plus 4.

01:57 So this is the demand expression let me simplify this order.

02:06 Okay when we simplify this we obtain that the price expression for this is minus 0 .005 times x then this plus 6 .4.

02:19 Okay now remember that the question is the revenue we need to minimize the revenue maximize the revenue sorry so the revenue is the price times x so that will give us minus 0 .005 times x squared and this plus 6 .4 times x.

02:35 Once we have this we derive this with respect to x in order to maximize this function so that will give us minus 0 .01 times x then this plus 6 .4 then we set this equal to 0 so solving for x that will be then 6 .4 divided by 0 .01.

02:55 So let's use our calculator for this and the value that we obtain is 640...

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