Suppose total benefits and total costs are given by by 100y...

Suppose total benefits and total costs are given by B(Y) = 100Y − 5Y^2 and C(Y) = 10Y^2. Also marginal benefits and marginal costs are given by MB(Y) = 100 - 10Y and MC(Y) = 20Y. What level of Y will yield the maximum net benefits?

Transcript

00:01 Hello students, we are given a question here where the total benefit is given and the total cost is also given.

00:07 We need to determine the level of y where the maximum benefit is yield.

00:12 So here we are supposed to know that what we are given the total benefit, by is equal to 100 y minus 8 y square.

00:22 We can write it as it as total benefit.

00:25 It means t b.

00:26 And a total cost tc is given as 10.

00:30 Times of y so here we are supposed to know that what we need to do we need to differentiate both of them so differentiation of t b can be written as mb marginal benefit okay students marginal benefit is d of t b by d y so now we can calculate mb marginal benefit is equal to it's a derivative with respect to y will be 100 minus a to 16 so it will be 16 y okay and here we are supposed to know that the differentiation of a total cost is nothing but equal to the marginal cost.

01:06 So marginal cost can be written as d of pc by d .y.

01:12 Now we can write here that the mc is equal to its derivative will be simple 10.

01:18 Now for here we are supposed to know that for optimum quantity, okay, for optimum quantity, what we need to do we need to equate both of them.

01:32 Like mb marginal benefit have to be equal to the marginal cost so marginal benefit is 100 minus 16 y is equal to here it is given as 10 marginal cost is 10.

01:46 Okay students so 100 minus 10 it means 90 is equal to 16 y so basically at that we can say that y is equal to 19 divided by 16 so when we divide this term by 2 we will get it as 45 and divided by 16.

02:03 16 divided by 2 which is 8 okay now we can put here we can say that putting it a putting ater in a total benefit and the total cost so total benefit it should be equal to tbs given as 100 y minus 8 by square so hundred and times y 45 divided by 8 minus 8 y square it means 45 divided by 8 whole square we can check again hundred y minus 8 y square okay so basically it should be equal to it is when we divided by 8 what we will get or rather than solving it by dividing we can write it as 45 hundred divided by 8 minus it should be equal to 45 square okay students so 45 square is 20 25 and divided by 1 8 will be cancelled by 1 8 it should be 8 simple now we can take a lcm so 4 ,500 minus 2 ,025.

03:09 We will get it as 2475 divided by 8...

Question

Suppose total benefits and total costs are given by B(Y) = 100Y − 5Y^2 and C(Y) = 10Y^2. Also marginal benefits and marginal costs are given by MB(Y) = 100 - 10Y and MC(Y) = 20Y. What level of Y will yield the maximum net benefits?

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