Access Millions of academic & study documents

Differential equations example problem 3

Content type
User Generated
Showing Page:
1/16

Sign up to view the full document!

lock_open Sign Up
Showing Page:
2/16

Sign up to view the full document!

lock_open Sign Up
Showing Page:
3/16

Sign up to view the full document!

lock_open Sign Up
End of Preview - Want to read all 16 pages?
Access Now
Unformatted Attachment Preview
Linear Models Now, for ek ek = 2 The, the models becomes P = P0 (2t ) Now, for t when P = 3P0 3P0 = P0 (2t ) Now, t = 1.58 hr Example 2. Hal-life of Plutonium A breeder reactor converts relatively stable uranium 238 into isotope plutonium 239. After 15 years it is determined that 0.043% of the initial amount A0 of plutonium has disintegrated. Find the half-life of this isotope if the rate of disintegration is proportional to the amount remaining. Solution: Let A(t) be the remaining amount, dA = kA dt A(0) = A0 Solving for A, A = cekt For c, A = A0 ek·0 , c = A0 Hence, A = A0 ekt At t = 15 years, A = (1 0.043%)A0 = 0.99957A0 . Now, for ek , 0.99957A0 = A0 ek·15 1 ek = 0.99957 15 Now, the equation becomes To solve for t when A = 0.5A0 , ⇣ ⌘ t A = A0 0.99957 15 ⇣ 0.5A0 = A0 0.99957 t = 24, 174 yrs t 15 ⌘ Example 3. Age of a Fossil A fossilized bone is found to contain one-thousandth of the C-14 level found in living matter. Estimate the age of the fossil. (Half-life of C-14 is 5, 600 years) Solution: Carbon dating also uses the model dA = kA dt Hence, A = A0 ekt as solved in the earlier examples. For ek , at t = 5, 600 years, A = 0.5A0 0.5A0 = A0 ek(5600) Hence, 1 ek = 0.5 5600 Now, the equation becomes For t at A = ⇣ ⌘ t A = A0 0.5 5600 A0 1000 ⇣ ⌘ t A0 5600 = A0 0.5 1000 t = 55, 808 yrs Newton’s Law of Cooling/Warming According to Newton’s empirical law of cooling/warming, the rate at which a temperature of a body is changing is di ...
Purchase document to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.
Studypool
4.7
Indeed
4.5
Sitejabber
4.4