Suppose that the revenue (In dollars) from the sale of product Is given by 70x 0.9x2 -003x8 How last Is the margina revenue changing when where * Is the number 0f u...

REVENUE The revenue $R$ (in dollars) generated by the sale of $x$ units of a digital camera is given by

$(x-135)^2 = -\dfrac{5}{7}(R-25,515)$.

Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue.

Were given that the revenue varies directly with the gallons sold. And we're told that we have a revenue of 47 40 When 12 gallons are sold. So we're going to use that information to come up with an equation. Then after we find the equation, we're going to find the revenue when 10.5 gallons are sold. So our direct various equation is going to be our equals Some constant variation or proportionality times 12. Oops, Not Times 12 yet times G. So now we replace what we have, so 47 40 Is equal to K, which we don't know, times 12 We're going to divide both sides by 12 47 40, divided by 12 comes out to be 3.95 is equal to K. Sorry, equation becomes R is equal to 3.95 G. So now that we have The equation, we replaced 10.54 G. So we get 3.95 Times 10.5. So this comes out to be $41.44. which would round up to be $41.48

So in this related rates question, we are given a situation where we know how our revenue and our coasts depend on our daily production. X And we're basically asked to find change in revenue James and cults and changing profit given a specific the introduction and a, um, they'll be changing production. So for eight, we are asked for the, uh, rate all change for revenue. So basically, we're asked to find PR P T. Now, this isn't even related quick rates. Question. This is just a derivative question. Um, so to find the rt t, we have to take time derivative off a t x minus 0.25 x where, So let's do that. Eso es t x becomes baby the ex The key minus the two comes down So we get zero point five times X and then by the chain rule, we could this other the ex d. T. So, by the way, um, that fact that the planner of the ex becomes the extra cheese also chamber. So the derivative off just X is one. But then because we had chain rule, it becomes one times dx DT, or just the ex CTU for short. So this is r D r D t. This was our basically are differentiated step then. Now we have to substitute and soul steps, which in this case, it's just substitution because the equation is where they solved. So it gets 80 times five, which is 400 minus half, uh, 4.5 times 50 times 100 which is 50 times five, which is the X t t. So the answer will become 100 and 50. Uh, if you work this out. So that's fairly straightforward. And B is really the same. It is the sea, the tea, So finding destructiveness, you can easier since there's only one x so derivative is seven the ex d. T. So subbing in quote unquote scolding, we find that the rates change off production costs starting five. So in this case, we can already tell that increasing the production is good for this company because the revenue is changing. My 150 all the costs is only changing by 35. So the five increase in production is improving our profits. Which is exactly what questions see is about. Um, it asks us to find the change off profit over time. Um so the easiest way to do this is to realize that profits is revenue minus costs. So the change in profit over time is the change in revenue over time, minus the change in costs overtime. So this is the differentiate step. Offer knitted rates. Now it's a bin. So this is 100 and 50 minus five or 100 and 15. So, as I mentioned, increasing the production for this company is currently increasing profits spot quite bits.

According to the ocean, we have given values C X keys closed and our necks is. But even you, we have given conditions. In this question. We can say that the value of C X is 105 x plus 70 1000. This is our first question on We also gives the value phonics. You know, collusion did is 2 45 x. This is a second equation. We it waiting board equation equation. We have C X is equal to biotics. We put the value we have 105 x plus 70. Housing is equal to 2 45 x all in solidity 2 45 x minus 100 fine X is equal to 70 1000 unfolding. We have 140 x is equal to 70 housing the value of X comes out Theories 500. We put no value. Oh, thanks. In c x Closed Daddy's 100 fight into 500 less 70 1000 There is a photo 1 22 under There is Toller When the soul the I'm my own uh, the profectus function be X is the your but even you Thanks. So right by though closed c x on the living we have. P X is equal to next minus C X. When you subtract value here we have the X. Is it going to 2 45 x minus 105 x close. So Andy cows and on solving that is 2 45 x minus 105 X minus. So in deposing the value comes out 100 fuller X minus. So Andy 1000 we can say that the value B X is one under food X minus 70,000. We can't the Value X is a core to 500 and the value of course, is 1 22 $500 and the value be X is 100 for drinks minus so indeed 1000.

The following problem we're going to be considering revenue from sales. And the first we have a owner of photocopy store, it's going to charge seven cents per copy for the 1st 100 copies and four cents per copy for each copy exceeding 100. So we have As 0.077 cents per copy. And that's from zero copies to 100. And then we know that after 100 copies you will have paid $7. So it's going to be seven plus and now it's only four cents X. And that's gonna be When X. is greater than 100 copies. So it can look like this which will result in this um function here where we see that A less resulted is $7. And then Um it's going to be .4 times Um X -100. So now we have a connected function where the price is going to decrease now her copy.