00:01
Hello, the objective of the question is to find the revenue function.
00:06
We have the rate of change of price given by p -x -x -equals to minus 250x divided by 16 plus xx2x2 plus x squared raised to the part 3 by 2.
00:24
Now for the revenue function, the formula is it is equals to price, price, price times the quantity.
00:36
So first we find the expression for the price.
00:40
For this, we integrate the p -x.
00:44
So we have integration p -x d -x is equals to integration of minus 250x divided by 16 plus x square raised to the power 3xxxxxxxxx plus x squared equals to t this implies that 2x dx equals to d t now derivative and integration cancel out each other so here i'm left with px is equal to 250 by 2 1 upon t raised to the power 3 by 2 d t.
01:41
Now integrate the expression and the value obtained is 250 by root t plus k, where k is the integrating constant.
01:59
Now according to the condition given in the question, we have when x.
02:04
Equals to 3 p of x equals to 50 so applying this condition to finding the integrating constant we have p of x as 250 replace the value of t which was 16 plus x squared plus k now substitute the value...