Problem 58 - College Algebra

Question

The daily demand and supply models for a carrot cake supplied by a bakery to a convenience store are given by the demand model \(N=40-3 p\) and the supply model \(N=\frac{p^{2}}{10},\) in which \(p\) is the price of the cake and \(N\) is the number of cakes sold or supplied each day to the convenience store. Explain how to determine the price at which supply and demand are equal. Then describe how to find how many carrot cakes can be supplied and sold each day at this price.

Step-by-Step Solution

Answer

Firstly, you need to set the demand and supply equations equal to each other and then solve for \(p\) which is the price of the cake. After finding \(p\), plug that into either the demand or supply equation to get the number of cakes \(N\). Solving this you will get the number of cakes that can be supplied and sold each day.

Step 1: Setting up the Equations

The first step is to take the demand model \(N=40-3 p\) and the supply model \(N=\frac{p^{2}}{10}\), and to set them equal to each other, because we want to find the price at which supply and demand are equal.

Step 2: Solving the Equations

Next, solve the equation \(40-3 p=\frac{p^{2}}{10}\). This is a quadratic equation, so it can be worked out by first multiplying the entire equation by 10 to get rid of the fraction, which gives \(400 - 30p = p^2\). Then, rearrange the equation to give all terms on one side of the equation: \( p^2 + 30p - 400 = 0\). Finally, solve this equation using the quadratic formula \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) in which a, b and c are the coefficients of the quadratic equation.

Step 3: Calculating number of Cakes

After calculating \(p\), plug this value back into either of the original demand or supply equations to calculate \(N\), the number of carrot cakes that can be supplied and sold each day at this price.