What questions were indeed, uh, given the art of the clerk computer firm, uh, markets. Two kinds of calculators, even and P two other price of those calculators, while Cuban and Q two other types of the calculators and party we gotta find the revenue functions so revenue function would be there price times quantity. So that will be given Cuban. Plus, we took you to Let's Substitute Q one and Q two over here and multiply with Even so, we have 78 p one minus six. Be one square minus three p one p two plus 66 b two minus three b one B two, minus six b two square If you simplify this further, we have 78 be one plus 66 p to minus six p. One square minus six B, two square minus six B one p two. So this is the revenue function, which depends on P one as well as speed. This is part A and part B. We gotta find the prices which will maximize the revenue. So we find are the one that is differentiating the complete equation with respect the P one. So we have 78 y P Tuesday is treated as a concert. So this is what will have an r p one on. We do the same procedure with P two so R p two would be 66 minus 12 p to minus six. B one. They quit both these 2 to 0. So we have 78 minus 12, even minus six. No, uh, P two is equal to zero. Let's divide both sides by six. So we have 13. Uh, maybe we can write 13 on the other side of the equation. So we have two p one plus p two was 13. The CIS look squalid equation one on if he quit this to zero as in 66 minus 12 p to minus six p. 10 This means that if you divide both sides were six again. So we have to be two plus p one as 11. Let's call it equation toe from equation one. We have the value off be, too, in terms off, even as 13 minus to be one. We substituted that over here, so we have two times 13 minus two B one plus B one is equal to 11. This is 26 minus three. P one is equal to 11, which means that three p one is 15, which means that the one is five. This is the value of P one, and let's substitute even over here. So we have P two as 30 my understand, which is just three. So the critical point is, uh, three and five is five and three on. Since this is the only critical point, there's no there's no significance off finding whether it has a local maximum minimum because there is only critical point and we're asking about maximizing. So it has to be the maximum eso these other two final answers for Cuban and Cuban in part C. We're supposed to find the quantity so Cuban would be replaced the values of P one and P two years. So P one is five will be to us three. So we have 78 minus 30 minus nine. So this week amount is 78 minus 39 which is nothing about 39. So this was the quantity off calculator cube one on. If you talk about you toe exactly the same procedure because we got a substitute these over here, so it will, it will be 66 minus three p one. Again, the one has five minus six p two Y p tours, three sort of 66 minus 15 minus 18, which is nothing but 66 minus 23 66 minus 33 which comes out as 33. So that's a skewed to which Waas asked on. The last point is, they're asking about the maximum revenue. So we're supposed to find the value of R P. One p two. So from here we take the equation, the function for our even p two. It's over here and we substitute P one and P two. So we have our even be, too as 78 the one where he even is. Let's write it again in the side, even as five and we two is three, so we have 78. In fact, there is no point of going in that details because we already have even Cuban P to Q two. So let's do it even Cuban plus p. Tokyo to because we already have all these four. We one is five you one is 39. B two is three. Q two is 33 So if you simplify this, we have 39 times five. We have 39 times five is 1 95 and 33 times three is 1990. We are both. We have the final answer as the final revenue in factors to 94. And since this is to the thousands, so we'll add 1000 over here. So this is the final or maximum revenue. And since the prices into the nearest 10, so we can say that this is the value of people in P two. But the price would be price. P one would be 5 50 on the prize. P two would be 30. This is what we can right here. And the quantity is to the nearest 100. So we'll say the quantity Cuban will be 3900 units on quantity. Q two will be 3300 units. That's shift a little bit of what down. So we have que tu as 3300 units. So these are the actual units. While he wanted you toe for the sake of putting in the equation. So all these would be our final on cells