00:01
Hello, so here we have our function, x is equal to 10 ,000 t squared over 9 plus t squared.
00:08
We have the annual sales at time t.
00:11
So then the rate of sales, r of t, is going to be equal to, well, dx, dt, the derivative of x, with respect to t.
00:21
We use the quotient rule here, and we have, well, 180 ,000 t plus 200 ,000 t cubed, minus or 20 ,000 t cubed minus 20 ,000 t cubed over 9 plus t squared squared, gives us 180 ,000 t over the quantity 9 plus t squared quantity squared.
00:51
Then we find the first derivative here, so it's going to be r prime of t, which is going to be equal to, again, using the quotient rule.
01:01
We get that we have our part of t is going to be 180 ,000 times the quantity 9 plus t squared times the quantity 9 minus 3t squared all divided by again the quantity 9 plus t squared squared okay then um when we set this equal to zero we do our prime of t equal to zero we do our prime of t equal to that implies that just 9 minus 3t squared is that equal to 0.
01:38
So now this implies that just 9 is equal to 3t squared.
01:43
And dividing through here by 3 gives us that t squared is going to be equal to, well, 9 over 3, which is 3.
01:51
So therefore, we have that t is going to be equal to, well, plus or minus square to 3, but t cannot be negative...