In a free-enterprise (supply and demand) economy, the amount...

In a free-enterprise (supply and demand) economy, the amount buyers are willing to pay for an item and the number of these items manufacturers are willing to produce depend on the price of the item. As the price increases, demand for the item decreases since buyers are less willing to pay the higher price. On the other hand, an increase in price increases the supply of the item since manufacturers are now more willing to supply it. When the supply and demand curves are graphed, their point of intersection is called the market equilibrium for the item. Suppose the monthly market demand $D$ (in tenthousands of gallons) for a new synthetic oil is related to the price $P$ in dollars by the equation $10 P^{2}+6 D=144 .$ For the market price $P,$ assume the amount $D$ that manufacturers are willing to supply is modeled by $8 P^{2}-8 P-4 D=12$ (a) What is the minimum price at which manufacturers are willing to begin supplying the oil? (b) Use this information to create a system of nonlinear equations, then solve the system to find the market equilibrium price (per gallon) and the quantity of oil supplied and sold at this price.

In a free-enterprise (supply and demand) economy, the amount buyers are willing to pay for an item and the number of these items manufacturers are willing to produce depend on the price of the item. As the price increases, demand for the item decreases since buyers are less willing to pay the higher price. On the other hand, an increase in price increases the supply of the item since manufacturers are now more willing to supply it. When the supply and demand curves are graphed, their point of intersection is called the market equilibrium for the item.
Suppose the monthly market demand $D$ (in tenthousands of gallons) for a new synthetic oil is related to the price $P$ in dollars by the equation $10 P^{2}+6 D=144 .$ For the market price $P,$ assume the amount $D$ that manufacturers are willing to supply is modeled by $8 P^{2}-8 P-4 D=12$
(a) What is the minimum price at which manufacturers are willing to begin supplying the oil? (b) Use this information to create a system of nonlinear equations, then solve the system to find the market equilibrium price (per gallon) and the quantity of oil supplied and sold at this price.

Key Concepts

Supply and Demand

This concept explains the relationship between how much producers are willing to supply at different prices and how much consumers are willing to purchase. As prices increase, consumer demand tends to decrease while producer supply tends to increase, forming the basis for analyzing market behavior in economics.

Market Equilibrium

Market equilibrium is reached when the quantity supplied by producers equals the quantity demanded by consumers. This intersection point on the supply and demand curves determines the prevailing market price and quantity, reflecting a state where there is no inherent pressure for change.

Systems of Nonlinear Equations

In economic modeling, particularly when relationships between quantities and prices are represented by quadratic or other nonlinear expressions, the resulting equations form a system of nonlinear equations. Solving these systems allows one to simultaneously determine unknown values such as equilibrium price and quantity.

Economic Modeling

Economic modeling involves using mathematical equations to represent the behaviors of supply, demand, and other economic variables. This approach helps in predicting market outcomes by capturing the dynamic interactions between various economic forces.

Transcript

00:01 So this is a multi -part problem and we're thinking about the supply and demand and how they both depend on price.

00:09 So we're given these two equations and they both relate these variables p and d.

00:14 In this first equation, this is about the demand and demand is based on price.

00:19 So d stands for demand and that'll change based on the price going up or down.

00:24 In the second equation, d is standing for supply and p is the price.

00:29 So suppliers will only make a certain amount if they have a certain price that they're working with.

00:36 And d means two things, but it's given by the same variable because it actually will mean the same thing when we get to part b of this, because we'll be looking for an equilibrium point.

00:47 So first, though, we have to solve for when supply is going to be greater than zero.

00:54 So we have to find what's the minimum price at which manufacturers are going to start supplying.

01:00 Oil.

01:01 If the price isn't high enough, there's not going to be any supply because they won't be making it out of money.

01:06 So we need to take the second equation and we need to find the price p that gets d higher than zero.

01:15 So to do that, we are going to set d equal to zero and solve and then we'll round up to the next to nearest cent.

01:23 So we'll set d equal to zero.

01:27 So we're doing 8p squared minus 8p minus 0, so we don't need to rate it equals 12.

01:36 And so we have a quadratic that we got to solve.

01:39 So 8p squared minus 8p minus 12 equals 0.

01:44 We subtract 12, so we can factor, if we can factor, or we can have our a, b, and c ready to go into the quadratic formula.

01:53 This does not factor, so we're going to use the quadratic formula.

01:56 X equals 8 plus or minus the square root of negative 8 squared which is 64 minus 4 times 8 times negative 12 and that's all over 16 and so we'll use a calculator for this because it's not rational we don't need a graph and calculator just a basic calculator we'll do so we get x equals 8 plus or minus the square root of 448 and so we might have up to two solutions here because we have the plus or minus option.

02:33 So the first option is x equals 1 .82 for the plus option and then the minus option is x equals negative 0 .822.

02:49 We're going to reject this solution because this is a negative price and that's not in the domain of this problem.

02:57 So though it is a solution to the quadratic, it's not actually.

02:59 A solution to this real world application -based problem.

03:03 So this is not a solution that's going to work for us.

03:06 This, however, is.

03:08 And so this is saying at $1 .82, they're not going to be making anything.

03:15 But if we raised this, if we rounded up to the next nearest cents, so $1 .83, that's when demand starts.

03:25 So this is the lowest minimum price for oil that menu.

03:29 Manufacturers will start making.

03:32 Okay, so that's the solution to our first part a.

03:35 We've got the price, the minimum price, that oil production will start at.

03:40 Now let's move on to part b.

03:44 So in part b, we're asked to use a system to find the equilibrium point.

03:49 And the system we're using is the system of two equations that's given at the beginning.

03:54 Nothing's different except for d means the same thing.

03:58 D means both supply and demand, but equilibrium is when supply equals demand.

04:05 So now d is just one variable.

04:07 It just stands for oil, what's the unit here? it stands for 10 ,000 of gallons of oil in the marketplace.

04:15 It's the same variable.

04:17 So we have to go about solving the system now for both p and d.

04:22 We need to find the price at which we'll have equilibrium, and also how much oil will be in the marketplace at that point.

04:29 Point.

04:31 So to solve this system, we can go about it a few different ways...

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