For a substance undergoing radioactive decay, the mass $\backslash (\backslash$boldsymbol$\{$m$\}($t$)\backslash )$ that persists after $\backslash ($ t $\backslash )$ years is an exponential function which we can write in the form

$\backslash [$

m$($t$)=$A e$^\{-$k t$\}$

$\backslash ]$

where $\backslash ($ t$=0\backslash )$ corresponds to your initial sample.

Radium$-226$ has a half$-$life of $1590$ years. The initial sample is $300$ mg $.$

$($i$)$ What are the values of $\backslash ($ A $\backslash )$ and of $\backslash (\backslash$boldsymbol$\{$k$\}\backslash )$ in the formula for $\backslash (\backslash$boldsymbol$\{$m$\}($t$)\backslash )$ above?

$\backslash ($ A$=\backslash )$

$\backslash [$

k$=$

$\backslash ]$

FORMATTING: Give exact answers, not decimal approximations. To write $\backslash (\backslash$ln $($a$)\backslash )$ type $\backslash (\backslash$ln $($a$)\backslash ).$

Determine the mass $\backslash ($ m$($t$)\backslash )$ that persists after $\backslash ($ t $\backslash )$ years.

$\backslash [$

m$($t$)=$

$\backslash ]$

$3$ mg $.$

FORMATTING:Give an exact formula $($do not round exponents$).$

$($ii$)$ What is the mass of Radium$-226$ after $2,000$ years?

Mass $=$ mg $.$

FORMATTING:Round the answer to the nearest integer.

$($iii$)$ When will the mass be reduced to $40$ mg $?$

Answer: pears.

FORMATTING:Round the answer to the nearest integer.