Calculus question, difficult.?

First you have to solve the differential equation C' = -kC:

dC/dt = -kC

1/C dC = -k dt

∫ 1/C dC = -k ∫ dt

ln|C| = -kt + D ... ... ... (D is the constant of integration)

C = e^(-kt + D)

C = C(t) = D e^(-kt)

So, if C is the amount of carbon-14 in a given sample, then C(t) is the amount of C-14 at some time t. Note that when t = 0, you have C(0) = D e° = D, so D is the initial amount of C-14.

You're given that the half-life of C-14 is 5588 years. This means that at time t = 5588, you have half of what you started with at time t = 0. That is,

C/D = 1/2 = e^(-5588k)

ln(1/2) = -5588k

k = -ln(1/2)/5588 ≈ 0.000124

Now for the Shroud problem: After some time t (the t you have to find), you have 91% of the C-14 remaining. So,

91% = 0.91 = e^(-kt)

ln(0.91) = -kt

t = -ln(0.91)/k ≈ 760.312 years

Note that this is the age of the artifact in 1988. Now, it would ≈ 785 years. It's not readily apparent which answer the question's looking for.

a)...dC/dt = -kC

So dC/C = -kdt

lnC = -kt + D where D is the constant of integration

C = e^(-kt+D) = e^D * e^(-kt)

At t= 0, C = e^D = C₀ since e^(-k0) = 1

So C = C₀e^(-kt)

When t = 5588 years, C/C₀ = 0.5. We can solve for k

C/C₀ = 0.5 = e^(-5588k)

ln(0.5) = -5588k

k = 1.2404 * 10^(-4) <=== Ans

b) In 1988, C/C₀ = 0.91

Thus 0.91 = e^(-kt). Solve for t with the k found above

t = ln(0.91)/(-k) = 760.3 ≈ 760 years is the age of the Shroud of Turin. <=== Ans

Hope this helps.

remark #a million. They wait till college to instruct it, and then they take 3 semesters to instruct it - 4 semesters in the event that they combine it with differential equations. remark #2. people who're majoring in basket-weaving infrequently take calculus. Neither do historical past majors or actual training majors. So, what do you return up with once you do the mathematics? like another concern, calculus isn't incredibly confusing as quickly as you comprehend a thank you to do it. once you hit delta-epsilon proofs, nonetheless (many times on the top of the 1st semester), you're anticipated to squeal like a caught pig. they are no longer relaxing in any respect.