00:01
So here we're giving the revenue function, r of t is equal to 63 t minus t squared over t squared plus 63 and we are to find where the maximum revenue occurs and also the maximum revenue.
00:19
Now we know that at maximum revenue, r prime of t will be equal to zero and therefore our first job here is to find the first derivative.
00:30
Now this is a quotient, so we are going to use the quotient rule and we'll call the numerator of u, the denominator v, our u prime, which is the first derivative of the numerator will be 63 minus 2 t and v prime is equal to 2t.
00:51
Now applying the caution rule, we say is v u prime minus u v v prime over v squared.
01:01
Here plugging in our values we're going to have let us create space here let me write this up here so we have our u prime is equal to 63 minus 2 t and our v prime is equal to 2 t and plugging those in we have our v t squared t squared multiplied by t squared plus 63 which is v multiplied by 63 minus 2 t which is our u prime subtract u 63 t minus t squared multiplied by 2 t which is our v prime divided by our v squared which is t squared plus 63 squared and equating this to zero since at maximum the first derivative is equal to 0.
02:03
Cross multiplying the denominator goes, we are left with the numerator and expanding the numerator there, we're going to have 63 t squared minus 2 t cubed plus 63 squared minus 123 minus 126 t and minus 126 t squared plus 2 t cubed and this is equal to zero.
02:33
And we'll see that this 2 t cubed will cancel out with this opposite sign 2 t cubed we subtract out and 63 t squared and also here we have 126 t squared so here we are going to have minus t squared so minus 63 t squared plus 63 t squared plus 63 squared minus 126 t is equal to 0 and dividing throughout by 63 positive 63 we get t squared minus 63 and plus 2 t which we can write as in the standard quadratic form so t squared plus 2t now, here, if we factorize this, we're going to have t minus 7, t plus 9, is equal to 0, and therefore t is equal to positive 7 or minus 9.
03:49
Now we throw out minus 9 and we just take positive 7 since t stands for time and it's from 0 to 63...