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

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 demand.

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're gonna be looking at this math question that's asking us to find the equilibrium demand using the following supply and demand equations where the supply equation is P is equal to the square root of 0.5 Q plus 16 and outside of the square root minus two. And the demand equation is P is equal to the square root of 36 minus 0.5 Q. Now the answer choice provided are a 40 B zero C 80 and D 25. Now, when looking at this question, I see that they're specifically asking for the equilibrium demand. So what does that mean? When do we reach equilibrium? Well, when it comes to supply and demand, we know that equilibrium is achieved when supply equals demand. So when you have a supply, that is the same as the demand, you reach a level of equilibrium. And to illustrate that we can set our supply and demand our supply equation equal to a demand equation. So let's do that. So we'll have the square root of 0.5 Q plus and outside of the square root, we have minus two and that'll be equal to the square root of 36 minus 0.5 Q. And now we can get to simplifying and solving for Q Q will represent the number of items uh for the demand at equilibrium in this case. So to get rid of the square roots on both sides, we have to square them. So on the left hand side, we have the square of 0.5 Q plus 16 and outside of that minus two and all of that squared, and that'll be equal to the square root of 36 minus 0.5 Q and all of that right hand side squared as well. Now we know that when we square a square root, it'll cancel out. So on the right hand side, when you cancel out the square root, since we have no other term, we're just left with what was inside the square root, which in this case was 36 minus 0.5 Q. Now, on the right hand, on the left hand side, we have a square root, but we also have a term outside of the square root. So we will cancel out the square root. However, we got to keep in mind that the minus two won't be canceled out. So it'll be part of the multiplication. So think about this as multiplying the square root of 0.5 Q plus and the minus two multiplying all that by itself. So we can't just drop the square root and keep what is inside of it. We have to keep in mind that we have a minus two outside of the square root. So once we square that entire term, we'll have 0.5 Q plus 16 minus for multiplying the square root of 0.5 Q plus 16 and outside of that square root plus four. Now, we can do some simplification. So we can move the 0.5 Q from the left hand side to the right hand side. And we can also move the plus 16 and the plus four from the left hand side to the right hand side. So by doing that, we'll be left with negative four, multiplying the square root of 0.5 Q plus 16 on the left hand side. And that'll be equal to, we had a 36 but now we are subtracting a 16 and a four from the left hand side. So subtracting 20 in total. So 36 minus 20 will be 16. And then we had a negative 0.5 Q and now we're also subtracting another 0.5 Q from the left hand side. So we'll have minus one Q. So in total, we have negative form multiplying the square root of 0.5 Q plus equal to 16 minus Q. Now, once again, we see that we have a square root and like we said before, to get rid of that, we have the square. So we'll have negative four, multiplying the screwed of 0.5 Q plus and all that squared. And what we do to the left hand side, we do to the right hand side. So on the right hand side, we'll have 16 minus Q and all that squared. Now, once we do the squaring, as we mentioned before, when you have a square root that is squared, the square root will cancel out. In this case, we have a negative four in front of the square root multiplying it. So once we do all that squaring, the negative four squared will be a positive 16. And that'll be multiplying what was inside of the square root, which now drops the square root. So it'll be 16 multiplying 0.5 Q plus 16. And that will be equal to when we square 16 minus Q, we will get 256 minus 32 Q plus Q squared. And now we can distribute on the left hand side. So 16 multiplied by 0.5, Q is the same as saying 16, divided by two, which is eight Q and then 16 multiplying a positive 16 will be plus 256. And that will continue to be equal to 256 minus 32 Q plus Q squared. And now we can set all this equal to zero, we can move everything to one side of the equation and set this equal to zero so that we have everything set up in a way that we can factor. So we have Q squared, that'll stay as a Q square because there are no other Q Q squared terms. So it'll be one Q squared. And then for the Q terms, we had a negative negative 32 Q on the right hand side and an eight Q on the left hand side. Now, if we move the A Q from the left hand side to the right hand side, we'll have a negative 32 Q minus A Q which is AM minus Q. And then we had positive 256 on the right hand side and the left hand side. So if we move it from the left hand side to the right hand side, we have 256 minus 256 which is zero. So we'll leave it as Q squared minus 40 Q equals zero. Now, we can factor out the Q from the Q squared and the negative 40 Q to have QQ multiplying Q minus 40. And that will continue to be equal to zero. So our two values for Q here would be Q equals zero or Q minus 40 equals zero or Q equals positive 40. Now, we can eliminate the Q equal zero because the number of items can't be zero for the demand. So we ignore Q equals zero and that will leave us with Q equals 40. And that'll be the equilibrium demand or answer choice A in this situation. 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 demand.

Key Concepts

Equilibrium in Supply and Demand

Solving Equations Involving Square Roots

Algebraic Manipulation and Equation Solving

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