They are essentially asking us to solve:
[tex] 357=900e^{-0.01155x} \implies\\
\frac{357}{900}=e^{-0.01155x} \implies\\
\ln{\frac{357}{900}}=-0.01155x \implies \\
x\approx 80.1 [/tex]
Solving this equation for x tells us that it will take around 80.1 years for the materials in the vault to reduce to 357 pounds.
Answer:
80.1 years of time will be needed to pass.
Step-by-step explanation:
Given expression:
[tex]A=A_o\times e^{-0.01155\times x}[/tex]
Given :[tex]A_o=900 pound[/tex]
A = 357 pounds
x = ?
Using the given expression:
[tex]A=A_o\times e^{-0.01155\times x}[/tex]
Taking ln both sides:
[tex]\ln A=\ln A_o -0.01155\times x[/tex]
[tex]\ln (357 pounds) =\ln (900) -0.01155\times x[/tex]
[tex]\ln (357 pounds) -\ln (900 pounds)=-0.01155\times x[/tex]
[tex]x=\ln \frac{900 pounds}{357 pounds}\times \frac{1}{0.01155}[/tex]
x = 80.0570 years ≈ 80.1 years
80.1 years of time will be needed to pass.
