Find the formula for f(t) and g(t)

Question

The half life of substance A is  19 years, and substance B decays at a rate of  30% each decade. 

Find a formula for a function f(t) that gives the amount of substance A, in milligrams, left after t years, given that the initial quantity was 100 milligrams.

f(t)=

 Find a formula for a function g(t) that gives the amount of substance B, in milligrams, left after t years, given that the initial quantity was 100 milligrams. 

g(t)=

Answer 1

Vanessa,

The formula is straightforward to write. To create the formula, you need to know:

* r= decay rate (in fraction lost per year)

* p = preservation rate (fraction kept per year) = 1-r

* t= number of years

* s= starting amount

Here’s how to think it out:

When the year starts, you have some amount s. By the end of the year, you’re going to lose rs. But how much is left? You’ll still have s-rs, or more compactly s(1-r). For simplicity, let’s give the name p to 1-r. So you’re left with sp.

Next year, you’ll start with sp and at the end you’ll have . sp*p. So at the end of 2 years, you’ll have sp².

so at the end of t years, you’ll have sp^t.

(Notice that s is not raised to the power, only the preservation rate is raised to the power).

in the problems you’ve been given, there are a couple of details to observe.

In the first problem, you know that

s = 100 mg.

But there’s a complication in the decay rate: they speak of “half life”, and they say the half life is 19 years. The half life is the amount of time it takes to reduce it by half.

in other words, if you knew the preservation rate p, you would write

p¹⁹= .5

You need to solve this for p, so that you can use the formula in years.

p = .5^-19

So in the first problem the formula is

f(x) = 100*(.5)^-19*t

now it’s your turn to make the second formula. Notice that we don’t have 19 years but 10 years to get to a decay rate of .3. So focus on a preservation rate of .7 in 10 years, and try to adjust the formula accordingly. If you have a question, please ask.