Was looking over my old Calculus I textbook and found this problem that I just can't solve.
The answers in the back of the book are 0.2 and ~0.18, respectively. I know this is a problem involving the chain rule and that (dp/dt)=20,000, area = (pi)r^2, and radius = 5. But I am stuck there. I am at a loss at how the answers are 0.2 and ~0.18, respectively.
The hardest parts of these are making the equations.
We know that dP/dt = 20000, and a = (pi)r^2 like you said. Now note that we have a relationship between P and a, specifically
P/a = 500,000 / (pi)5^2
Rearranging this equation gives a = (P (pi) 5^2 / 500000)
we can take the derivative w.r.t time here and we get...
da/dt = dP/dt ( (pi)5^2 / 500000) and we can plug in dP/dt, which was 20,000.
This tells us that da/dt = pi after calculating the above.
Now we need another equation with da/dt, but we do know the area equation, so we can take the derivative of it.
a = (pi) r^2 Take the derivative (Here is where the chain rule comes in)
da/dt = 2(pi)r (dr/dt), and we know r is 5.
Now combine both equations and we get (pi) = 2 (pi) (r) dr/dt
rearranging gives dr/dt = 1/2*r which is 0.1 when we plug in the r = 5.
Since the question asks for change of diameter, and we have change of radius, we just multiply by 2. This gives the answer of 0.2.
For the part about 5 years later, we need the new radius.
The new population is P`= 500,000 + 20,000 *5 = 600,000
Since we know that Population / area is a constant, we can use
500,000/(pi)5^2 = 600,000/(pi)r^2 The left side is the ratio from 5 years ago, where we knew the population was 500,000 and radius was 5. The right side is the new population of 600,000. Solving for r gives r = sqrt(30)
Using the relationship we already found, dr/dt = 1/(2r) we plus in the new r and see that the change of radius 5 years later is 0.091ish, so multiplying by 2 for the diameter gives the .18
= 10*f(n-1)+n ; f(0)=0 just to explore these numbers a bit. The square-root of the odd terms has these interesting strings of numbers, the same sequence of repetitions but varying in length, odd-term to odd-term. Anyway I just happened to be resizing a window and saw this triangle pop-out:
