Find the price p that will produce each of the following revenues. Round answers to two decimal

Find the price p that will produce each of the following...

Key Concepts

Rounding

Rounding involves adjusting a numerical result to a specified degree of precision, which is crucial in applied mathematics and real-world scenarios. In pricing contexts, rounding to two decimal places is standard to align prices with conventional currency formats.

Revenue Function

A revenue function represents the total income generated by selling a product at a particular price and is typically defined as the product of the price per unit and the quantity sold. It models how changes in price affect the revenue collected, often incorporating a demand function that describes how many units are sold at each price.

Quadratic Equations

Many revenue problems lead to quadratic equations when the demand function is linear, resulting in a quadratic relationship between price and revenue. Solving these equations helps determine the price that will yield a specific revenue, using methods like factoring, completing the square, or the quadratic formula.

Transcript

00:01 So we know that q stands for the number of hamburgs that will be sold, and that is controlled by the price in dollars.

00:17 And so we know that the revenue, in terms of the price, will equal the price times the number sold, and this is the number sold.

00:31 So we want to know in a, when will this model of p, and i'm already going to distribute it through, 1600p minus 200 p squared, when will that revenue equal $2 ,800? so we set it equal to zero, and we have 200 p squared minus 1 ,600 p plus 2 ,800.

01:03 And let's use, actually, let's divide everything by 200.

01:09 And when we divide everything by 200, we get p squared minus and 1 ,600 divided by 200.

01:17 Will end up giving us 8.

01:21 And then 2 ,800 divided by 200, we'll lose the 2 zeros, and then 2 goes into 2814 times.

01:29 And this, let's see, do i see if a factoring that's going to produce this? if i use the, i always like to try to factor if i can.

01:38 And if i use 2 and 7, that's not going to give me the right middle term.

01:43 And that's about 1 and 14.

01:44 It's definitely not going to work.

01:45 So let's go to quadratic formula.

01:47 So we have p is equal to 8 plus or minus the square root of 64 minus 4 times 1 times 14 all over 2a, which is 2.

02:00 And so we have that discriminant for 64 minus 4 times 14.

02:07 That discriminant is 8.

02:09 So this becomes 8 plus or minus the square root of 8.

02:14 And again, we're not, we want this in money.

02:17 I'm going to put this in my calculator and see what we get, 8 plus square root of 8.

02:25 And close my parentheses divided by 2.

02:28 The plus gives us, if we have a price of $5 .41, we'll get that...

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