Question

The half-life is a substance is 375 years. If 70 grams is present now, how much will be present in 500 years?

Use the formula pictured

Answer 1

Answer:

27.76 grams will be present in 500 years

Step-by-step explanation:

The given formula is $A=A_{o}e^{kt}$ , where A is the value of the substance in t years, and $A_{o}$ is the initial value

∵ The half-life is a substance is 375 years

- Substitute A by $\frac{1}{2}A_{o}$ and t by 375 to find the value of k

∴ $\frac{1}{2}A_{o}=A_{o}e^{375k}$

- Divide both sides by $A_{o}$

∴ $\frac{1}{2}=e^{375k}$

- Insert ㏑ in both sides

∴ ㏑( $\frac{1}{2}$ ) = ㏑ ( $e^{375k}$ )

- Remember ㏑ ( $e^{n}$ ) = n

∵ ㏑ ( $e^{375k}$ ) = 375 k

∴ ㏑( $\frac{1}{2}$ ) = 375 k

- Divide both sides by 375

∴ k ≈ -0.00185

∴  $A=A_{o}e^{-0.00185t}$

∵ 70 grams is present now

- That means the initial value is 70 grams

∴ $A_{o}$ = 70

∵ The time is 500 years

∴ t = 500

- Substitute the values of $A_{o}$ and t in the formula

∵ $A=70e^{-0.00185(500)}$

∴ A = 27.76

∴ 27.76 grams will be present in 500 years