3) A sample of 42 grams of an unknown substance has a half-life of 1,300 years. (a) Write an equation to determine the amount of substance, S, left after t years.

(b) Approximately how long will it take for 0.2 grams of substance to remain (to the nearest year)?

a)

b) It will take 10,692 years.

Step-by-step explanation:

Exponential function to determine the amount of substance:

An exponential function to determine the amout of substance after t years is given by:

In which A(0) is the initial amount and r is the decay rate, as a decimal.

(a) Write an equation to determine the amount of substance, S, left after t years.

Half-life of 1300 years means that .

We use this to find r. So

Sample of 42 grams means that . So

Replacing A by S, just for notation purposes

(b) Approximately how long will it take for 0.2 grams of substance to remain (to the nearest year)?

This is t when . So

It will take 10,692 years.