$\"\"$

The half-life of Rubidium- $\"\"$ is $\"\"$ billion years.

Find the value of $\"\"$ and the equation of decay.

Let $\"\"$ is a intial amount of substance, then the amount $\"\"$ that remains after $\"\"$ billion years can be represents the $\"\"$ .

$\"\"$ (Exponential decay formula)

$\"\"$ (Substitute $\"\"$ and $\"\"$ )

$\"\"$ (Divide each side by $\"\"$ )

$\"\"$ (Cancel common terms)

$\"\"$ (Take ln on each side)

$\"\"$

Divide each side by $\"\"$ .

$\"\"$

$\"\"$ (Cancel common terms)

$\"\"$ (Substitute: $\"\"$ )

$\"\"$ (Simplify)

$\"\"$ (Substitute $\"\"$ )

Thus, the equation for the decay of Rubidium- $\"\"$ is $\"\"$ .

$\"\"$

Find the how long will take the speciman to decay to only $\"\"$ milligrams of Rubidium- $\"\"$ .

Here $\"\"$ and $\"\"$ milligrams of Rubidium- $\"\"$ .

$\"\"$ (Substitute $\"\"$ and $\"\"$ )

$\"\"$ (Divide each side by $\"\"$ )

$\"\"$ (Cancel common terms)

$\"\"$ (Take ln on each side)

$\"\"$ ( $\"\"$ )

$\"\"$ (Substitute: $\"\"$ )

Divide each side by $\"\"$ .

$\"\"$

$\"\"$ (Cancel common terms)

It will take the specimen about $\"\"$ years to decay to only $\"\"$ milligrams of Rubidium- $\"\"$ .

$\"\"$

Find how many milligrams of Rubidium- $\"\"$ will be left after $\"\"$ million years.

Here $\"\"$ milligrams of Rubidium- $\"\"$ and $\"\"$ years.

$\"\"$ (Substitute $\"\"$ and $\"\"$ )

$\"\"$ (Multiply exponents)

$\"\"$ (Use calculator: $\"\"$ )

$\"\"$ (Simplify)

About $\"\"$ milligrams of Rubidium- $\"\"$ will be left after $\"\"$ million years.

$\"\"$

Find how long will take Rubidium- $\"\"$ to decay to one sixth of its original amount.

$\"\"$ (Substitute $\"\"$ )

$\"\"$ (Divide each side by $\"\"$ )

$\"\"$ (Cancel common terms)

$\"\"$ (Take ln on each side)

$\"\"$ ( $\"\"$ )

$\"\"$ (Substitute: $\"\"$ )

Divide each side by $\"\"$

$\"\"$

$\"\"$ (Cancel common terms)

It will take $\"\"$ years to decay to one sixth of its original amount.

$\"\"$

a. The equation for the decay of Rubidium- $\"\"$ is $\"\"$ .

b. It will take the specimen about $\"\"$ years to decay to only $\"\"$ milligrams of Rubidium- $\"\"$ .

c. $\"\"$ milligrams of Rubidium- $\"\"$ will be left after $\"\"$ million years.

d. It will take Rubidium- $\"\"$ $\"\"$ years to decay to one sixth of its original amount.