Refer to Problem 88 . Find the price p that will produce each of the following revenues. Round a

Refer to Problem 88 . Find the price p that will produce...

Key Concepts

Quadratic Equations

When the demand function is linear, the revenue function becomes a quadratic function of the price. Solving a given revenue target, therefore, often involves setting up a quadratic equation where the revenue function equals a constant value. Mastery of quadratic equations, including methods of factoring, using the quadratic formula, or completing the square, is crucial in navigating these problems.

Solving Equations and Rounding Approximations

Once the quadratic equation is formed from the revenue function, the next step is to solve for the price that meets the revenue target. This process requires careful algebraic manipulation and sometimes using approximation methods, such as decimal rounding to a specified number of places (e.g., rounding to two decimal places). Such rounding is important to present the answer in a practical, usable form.

Revenue Function

The revenue function relates the price at which a product is sold and the quantity of units sold to the total revenue generated. Typically, the function is modeled as the product of the price and the demand function (which describes how many units will be sold at a given price). This model is foundational in economics and business for analyzing how changes in price can affect earnings.

Transcript

00:01 Now we're talking about pizzas, and q is standing for the number of pizzas that will be sold for a given price.

00:08 And that is, oops, this is 800.

00:11 Looking at the wrong question.

00:14 That is going to be, i'm sorry, 8 ,000 minus 400 times p.

00:20 And so the revenue in terms of the price is going to be the price of this times the number sold, which will be 8 ,000.

00:31 Minus 400p and we want to know when will that revenue for part a when will the revenue equal 38 ,000 so when will it equal 38 ,000 and that's when let's distribute we have 8 ,000 minus 400 p squared that's when that's equal to it and getting everything on the left side we would have 400 p squared minus 8 ,000 p plus 38 ,000 and is equal to zero and just to make my numbers smaller.

01:12 Well, actually, because i'm going to keep changing this number, normally i would divide everything through here by, well, that's at least divided through by 100.

01:20 So we'll have 4p squared minus 80p and then 380.

01:27 So at least our numbers are a little bit smaller.

01:30 And again, i could divide everything through by 400.

01:33 And let's make sure i did that right.

01:35 Divide and take away two, tick away two, tick away two.

01:39 Looks good.

01:40 So p will equal 80 plus or minus square root of 80 squared minus 4 times 4 times 380 ,000, all over four times two times four which will be eight.

02:00 And now i'm just going to put this into my calculator.

02:03 Well, let's find the discriminant first.

02:05 80 squared minus four times four times 380.

02:11 That discriminant becomes 320.

02:15 So we have 80.

02:19 Left parentheses in my calculator, 80 plus plus square root of 320.

02:26 And my parentheses divided by 8.

02:29 And one price for the pizza would be $12 and basically $24.

02:35 And let's see what happens if we do a change that to a minus sign, a subtraction sign.

02:43 And that gives us an alternative price of $7 .76...