Question

A radioactive substance decays according to the formula w = 20e¡kt grams where t is the time in hours. a find k given that after 50 hours the weight is 10 grams. b find the weight of radioactive substance present at: i t = 0 hours ii t = 24 hours iii t = 1 week. c how long will it take for the weight to reach 1 gram? d find the rate of radioactive decay at: i t = 100 hours ii t = 1000 hours. e show that dw dt is proportional to the weight of substance remaining.

Answer 1

W=20 e(-kt)  
A. Rearranging gives k= -(ln(w/20)/t 
 Substituting w= 10 and solving gives k=0.014 
 B. Using W=20e(-kt). After 0 hours, W=20. After 24 hours, W=14.29g. After 1 week (24x7=168h) W=1.9g 
 C. Rearranging gives t=-(ln(10/20)/k. Substituting w=1 and solving gives t=214 hours. 
 D. Differentiating gives dW/ dt = -20ke(-kt). Solving for t=100 gives dW/dt = 0.07g/h. Solving for t=1000 gives 0.0000002g/h 
 E. dW/dt = -20ke(-kt). But W=20e(-kt) so dW/dt = -kW