Question 12 A manufacturing company produces and sells...

Question 12 A manufacturing company produces and sells televisions. The cost function is g C(x) = 2x + 3\sqrt{x} + 150 Find the marginal cost when 50 televisions are being produced. A. Decreasing by $0.89 per television B. Increasing by $2.21 per television C. Decreasing by $5.89 per television D. Increasing by $12.55 per television E. Increasing by $8.48 per television

Transcript

00:02 In this question, we are given that p of x is equal to negative x over 30 plus 300, where p is the selling price per tv, per tv, and we are given cx is equal to 30x plus 150 ,000, where x is number of tv, now the revenue equation rx is given by so we know that rx the revenue is the number of pieces produced by the by the price at which they shall be sold so this is x by px and we have our x by minus x over 30 plus 300 giving us the revenue equation as 300 x minus x squared over 30 and this is our revenue equation.

01:38 Next we want the break even points.

01:50 Now we know that breaking even occurs when the revenue equals the cost of production and this is assuming that there are no other cost related to producing or selling the product now our x is 300 x minus x squared over 30 is equal to the cx is 30x plus 150 ,000 and if we rearrange this we are going to get x squared minus 8100 8100 x plus 4 million 500 thousand is equal to zero now solving this using the the quadratic equation we are going to get where our a is one our b is negative 8100 and our c is 4 .5 million let's just write that in full we are going to get so that is million there is no comma there so using the quadratic equation we're going to get our values for x1 and 2 as 600 and 7500 and 500 and if we plug in r at 600 we will see that r at 600 is equal to 168 ,000 and c at 600 is the cost function.

03:48 We also get 168 ,000 and this proves that this is really the lower limit where we have our break -even.

04:00 So if pre -givine starts at 600 and at 7 ,500, so this region is our profit region and our loss regions will be at x is less than 600 and x is greater than 7 ,500.

04:35 This will be our loss region that's it and going forward we need to find the profit function so the profit profit function now we know that profit is rx minus cx and we here we call our profit t x t of x is equal to our profit t of x is equal to r of x minus c of x and r of x is 300 x minus x squared over 30 subtract our c of x is 30x plus 150000 in brackets and here we get our t of x as x squared negative x squared over 30 plus 270x minus 150 ,000.

06:11 And this is our profit function.

06:18 Now finally we are to find the marginal profit and our marginal profit function.

06:27 So we're going to try to fit it in here so that we don't delete anything.

06:35 Now we know that our marginal profit function...

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