The demand for a product is increasing exponentially (i.e.,...

The demand for a product is increasing exponentially (i.e., at a constant percentage rate.) The demand in 2015 was 8000 units/month, up from a demand of 5000 units/month in 1986. A) At what percentage rate is the demand growing? The demand is growing at a rate of % per year. B) When will the demand reach 12000 units/month? The demand will reach 12000 units/month in the year C) What will the demand be in the year 2031? The demand would be units/month in the year 2031.

Transcript

00:01 Hi, in the given problem we are given with the demand function of the commodity and we are required to find the expenditure change with respect to price when the price is 100.

00:12 So the price x is 110 dollars so at that price you have to find the change in the expenditure now here the demand function d x is great is equal to 6 ,000 e to the power minus 0 .07 x and the expenditure function is x times x x times d x which is x times 6 ,000 e to the power minus 0 .07 x now in order to find the change in the expenditure we will be first finding the derivative of this so this is e prime x this is equal to x times minus 420 e to the power minus 0 .07x plus 6 ,000 e to the power minus 0 .07 x now so this can be further written as 420e to the power minus 0 .07 x times 100 over 7 minus x.

01:13 So this is the required function.

01:15 I mean, there's a function for the rate of change in expenditure.

01:19 Now, we have to find the e prime at the part a, we have to find e prime 110.

01:26 So that is when x is equal to 110.

01:28 So this would become 420, e to the power minus.

01:31 0 .07 x times 100 over 7 minus 110 so this comes out to be minus 18 .20 so this is the rate this is a decreasing rate it means decreasing at 18 .2 dollars but 18 .2 per dollar so that's the that's the expenditure rate in the first part in the second part we are required to check if at what interval the expenditure x is decreasing so we have to check ex is decreasing in what interval.

02:07 So in order to find that, we know that it is decreasing when e prime x is less than 0.

02:16 So here this is less than 0.

02:18 It means this whole term is less than 0...

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