Respuesta :

Answer:

27.76 grams will be present in 500 years

Step-by-step explanation:

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

∵ The half-life is a substance is 375 years

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

∴ [tex]\frac{1}{2}A_{o}=A_{o}e^{375k}[/tex]

- Divide both sides by [tex]A_{o}[/tex]

∴ [tex]\frac{1}{2}=e^{375k}[/tex]

- Insert ㏑ in both sides

∴ ㏑( [tex]\frac{1}{2}[/tex] ) = ㏑ ( [tex]e^{375k}[/tex] )

- Remember ㏑ ( [tex]e^{n}[/tex] ) = n

∵ ㏑ ( [tex]e^{375k}[/tex] ) = 375 k

∴ ㏑( [tex]\frac{1}{2}[/tex] ) = 375 k

- Divide both sides by 375

∴ k ≈ -0.00185

∴  [tex]A=A_{o}e^{-0.00185t}[/tex]

∵ 70 grams is present now

- That means the initial value is 70 grams

∴ [tex]A_{o}[/tex] = 70

∵ The time is 500 years

∴ t = 500

- Substitute the values of [tex]A_{o}[/tex] and t in the formula

∵ [tex]A=70e^{-0.00185(500)}[/tex]

∴ A = 27.76

27.76 grams will be present in 500 years

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