Question

The radioactive substance uranium-240 has a half-life of 14 hours. The amount At of a sample of uranium-240 remaining (in grams) after t hours is given by the following exponential function.

Answer 1

Step 1

Given;

`A(t)=5600((1)/(2))^{(t)/(14)}`

Required; To find the amount A(t) of a sample of uranium-240 remaining (in grams) after 13 hours and 60 hours

Step 2

Find the initial amount. To do this we set t=0

`\begin{gathered} A(0)=5600((1)/(2)_)^{(0)/(14)} \\ A(0)=5600grams \end{gathered}`

Hence the equation remains valid

Step 3

Find the amount A(t) left after 13 hours

`A(13)=5600((1)/(2))^{(13)/(14)}`
`\begin{gathered} =5600\cdot \frac{1^{(13)/(14)}}{2^{(13)/(14)}} \\ =(5600)/(1)\cdot \frac{1}{2^{(13)/(14)}} \\ =\frac{5600}{2^{(13)/(14)}} \\ =\frac{2^5\cdot \:175}{2^{(13)/(14)}} \\ =2^{(57)/(14)}\cdot\:175=2942.11858 \\ =2942grams \end{gathered}`

Step 4

Find the amount A(t) left after 60 hours

Answers;

`\begin{gathered} A(13)=2942grams\text{ to the nearest gram} \\ A(60)=287grams\text{ to the nearest gram} \end{gathered}`