Question

I have two problems I need help on, can someone show and explain step by step how to do the problems?

For #1, how would you differentiate and solve the inital-value problem?

For #2, I don't understand what I am doing wrong with the problem..I am sure it is a decay problem(?) using the equation Y(t) = Yo * e^KT ..I kept on getting the incorrect answer..


1) 8sin^2(y) dx + 8cos^2(x) dy = 0 , y(π/4) = π/4

2) Wood deposits recovered from an archaeological site contain 40% of the carbon-14 they originally contained. How long ago did the tree from which the wood was obtained die? (The half-life of carbon C-14 is 5730 years. Round your answer to the nearest year.)

Answer 1

#1 you don't have to differentiate. You are given the differentials, and you need to integrate

8sin^2(y) dx + 8cos^2(x) dy = 0
cos^2(x) dy = - sin^2(y) dx
dy/sin^2(y) = -dx/cos^2(dx)
-csc^2(y) dy = sec^2(x) dx
now integrate:
cot(y) = tan(x) + c
now, since (π/4,π/4) is on the curve,
1 = 1 + c
so, c=0
cot(y) = tan(x)
y = arccot(tan(x))
except for the asymptotes of cot(x),
y = π/2 - x, since
tan(x) = cot(π/2-x)

for the wood problem, it's easy to work with base 2, since you have a half-life. The amount left after t years is

y(t) = (1/2)^(t/5730)
so, you want t where
(1/2)^(t/5730) = 0.40
(t/5730) ln(1/2) = ln(0.4)
t = 5730 ln(.4)/ln(.5) = 7574.6

If you really want to use base e, then show us your steps, and we can spot where you went wrong. Note that since e^ln2 = 2,
1/2 = e^(-ln2)
(1/2)^(t/5730) = e^((-ln2)*t/5730) = e^-.000121t