Substance A decomposes at a rate proportional to the amount of A present. a) Write an equation relating A to the amount left of an initial amount Upper A 0 after time t. b) It is found that 18 lb of A will reduce to 9 lb in 4.2 hr. After how long will there be only 1 lb left?

Step-by-step explanation:

a) The equation relating A to the amount left of an initial amount Upper A 0 after time t is

A (t) = A0e^ (kt)

b) It is found that 18 lb of A will reduce to 9 lb in 4.2 hr. It means that

A0 = 18

A = 9

t = 4.2

Therefore, substituting into the formula, it becomes

9 = 18e^ (4.2k)

9/18 = e^ (4.2k)

0.5 = e^ (4.2k)

Taking ln of both sides of the equation, it becomes

Ln0.5 = ln e^ (4.2k)

- 0.693 = 4.2k

k = - 0.693/4.2

k = - 0.165

The equation becomes

A (t) = 18e^ ( - 0.165t)

For A = 1 (1 lb left), then

1 = 18e^ ( - 0.165t)

1/18 = e^ ( - 0.165t)

0.056 = e^ ( - 0.165t)

Taking ln of both sides, it becomes

Ln0.056 = ln e^ ( - 0.165t)

- 2.88 = - 0.165t

t = - 2.88/-0.165

t = 17.5 hours

After 17.5 hours, 1 lb would be left