Hi,
First, let's figure out the value of x by plugging in p = 16 into our given function. After plugging in for p, you have probably realized that it is impossible to solve the polynomial by using regular factoring methods. Thus we can either use completing the square method or the quadratic formula.
When I used the quadratic formula, I got positive and negative values of x. I want you to ignore the negative value of x because supply cannot be negative. Why? Because can you have a negative of an item? The answer is no, lol. Anyways, now that we know the value of x, which should be around x = 18.9411, we need to figure out what is given to us in the question.
Well, just by looking at the question, I figured out that the value of p is 16, and we are also told that p is "increasing," meaning dp/dt or derivative of p(price) with respect to time is 0.76. I did this using dp/dt you could probably do it using other variables/letters.
Anyways, you would now have to use implicit differentiation to isolate dx/dt derivative of x(supply) with respect to time. You will also use "product rule" while doing your differentiation, so please be careful as it tends to throw many students off. You should get something like this:
dx/dt = [--dp/dt * (--x / (2 * (p)^(1/2))) -- 2p] / [(2x -- (p)^(1/2)]
Plug in all the given values of x, p, dp/dt and you should get dx/dt = 0.770889.
This means that the supply will change by 0.770889.
I hope this helps!
