Solve each problem. The supply and demand equations for a certain...

Question

Solve each problem. The supply and demand equations for a certain commodity are given.
supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q)

Find the equilibrium price (in dollars).

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're gonna be looking at this math question that states find the equilibrium price in dollars using the following supply and demand equations where the supply equation is P is equal to the square roots of 0.5 Q plus and then outside of the square root minus two. And the demand equation is P is equal to the square roots of minus 0.5 Q. So the answer to that they provide us are a $4 B $5 c $4. and D $6. Now, when looking at this question, we see that they are specifically asking for the equilibrium price. So we have to ask ourselves, when do we reach equilibrium? When we have supply and demand? Well, when it comes to supply and demand, we know that equilibrium will be achieved when supply is equal to demand. So when you have the same supply and demand, you will reach a level of equilibrium. So to find our answer, we can set our supply equation equal to our demand equation. To see the what the equilibrium is. So let's do that. So we'll have the squirt of 0.5 Q plus and then outside of the square root minus two, and that will be equal to our demand equation, which is the square root of minus 0.5 Q. Now we can start solving for Q. And to do that, we see that each side, the left hand side and the right hand side both have a square root and to get rid of the square root, we're gonna square both sides. So we'll have the square root of 0.5 Q plus 16. And outside of that minus two, we'll have that squared and that'll be equal to the square root of 36 minus 0.5 Q with that squared as well. Now, on the right hand side, we only have a square root and no other term. And we know with when you have a square root and you have that squared, you get rid of the square root. So on, on our right hand side, we'll be left with 36 minus 0.5 Q, which is exactly what was inside the square root, but we just got rid of the square root. Now, on the left hand side, we have a square root and a number term outside of the square root. So when we square this, we will get rid of the square root. But we also have to keep in mind that we're also squaring the minus two. So this is the same, think about it as multiplying the zero point, the square root of 0.5 Q plus 16 with the minus two outside multiplying all that by itself. So when you do that, you will get 0.5 Q plus 16, which is what was inside the square root. But now we'll also have minus four multiplying the square root of 0.5 Q plus 16 and outside of the square root A plus four. So now we can do some simplification. So we can bring over the plus four and the plus 16 from the left hand side to the right hand side. And we can also bring over the 0.5 Q from the left hand side to the right hand side. So once we do that, we'll be left with negative four, multiplying the square root of 0.5 Q plus 16 and that'll be equal to, we had 36 minus 0.5 Q and that, that 36 we're taking out we're subtracting and A four. So in total, we're gonna be subtracting a 20. So we'll have 16, 36 minus 20 and 16. And then we had a negative 0.5 Q and we're also subtracting another 0.5 Q. So we'll have minus Q A full Q. So it'll be negative four, multiplying the square root of 0.5 Q plus 16 equals 16 minus Q. Now we see that once again, we have a square root that we want to get rid of. So we're gonna square once again. So we'll have negative four, multiplying the square root of 0.5 Q plus 16. And we'll have that squared and whatever you do to the left hand side, we do to the right hand side. So we'll have 16 minus Q squared, the entirety of 16 minus Q squared. So once we do that, we will have on the left hand side, once we square that we'll have 16, multiplying 0.5 Q plus 16. So we got rid of the square root and we squared the negative four in front of the square root and that'll be equal to 60 minus Q squared, which is 256 minus 32 Q plus Q squared. Now let's do some distribution on the left hand side, we have a 16 that's multiplying the 0.5 Q plus 16. So let's distribute that. So we have 16 multiplying the 0.5 Q which is the same as saying 16, divided by two, which is eight Q and then we have 16 multiplying a positive which is plus 256. So we have eight Q plus 256 equals 256 minus 32 Q plus Q squared. So the right hand side stays the same for now. So now we can do some more simplification. Let's, we can bring over everything on the left hand side or everything on the right hand side, we can bring them all together on one side of the equation and said that equation equal to zero. So once we do that, we'll have the Q squared and the Q square stay the same because we have no other Q squared term, then we had a minus 32 Q. So a negative 32 Q and we had a positive 88 Q on the left hand side. If we bring that over to the right hand side, we'll have a minus 40 Q. And then we had positive 2 56 on the left hand side and a positive 2 56 on the right hand side. So once you move either or to the other side, we have a zero. So they cancel each other out. So we'll have Q squared minus 40 Q equals zero. And now it's just a matter of factoring to solve for Q. We see that both of these terms Q squared and minus 40 Q, we, they both have a Q in them. So we can factor out a Q. So we'll have Q multiplying Q minus inside of parentheses. So now we can solve for Q, we know that it's possible. Uh The outside Q, it could just be Q equal zero. And the other answer for Q will be Q minus 40 equals zero, N Q equals positive 40. Now, we can eliminate Q equal zero because we can't have a supply of zero or a demand of zero. So we have to eliminate zero. Therefore, our supply will be or our Q will be 40. And now with that 40 we can plug that back into our supply equation. So let's do that. So our supply equation was P equals the square root of 0.5, Q plus and minus two outside of the square root. So when we plug in 40 we'll have P is equal to the square root of 0.5, multiplied by 40 plus 16. And then outside of the square root, A minus two 0.5, multiplying A 40 is the same as saying 40 divided by two, which is 20. So we have the square root of 20 plus and then minus two outside 20 plus 16 is 36. So that's the square root of 36. So we'll have, which is six. So we'll have P equals six minus two or P equals four dollars. Keep sending for price. So we have that our price at equilibrium will be $ or answer choice. A in this case. So I hope that was helpful. And until next time.

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q) Find the equilibrium price (in dollars).

Key Concepts

Supply and Demand Equations

Equilibrium Price

Algebraic Manipulation

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