Question

Please help with this question!

Answer 1

The equation for how long it will take for Veronica's money to increase by a factor of 10 in terms of t and k is:

`\boxed{t=(1)/(k)\ln 10}`

Explanation:

To increase a number by a factor of 10 means to multiply the original number by 10.

Therefore, to find how long it will take for Veronica's money to increase by a factor of 10, given that the initial amount is A₀, substitute A(t) = 10A₀ into the given function and solve for t.

`\begin{aligned}&\textsf{Given function:} \quad &A(t)&=A_0e^(kt)\\&\textsf{Substitute $A(t)=10A_0$:} \qquad \qquad & 10A_0&=A_0e^(kt)\\&\textsf{Divide both sides by $A_0$:} \quad & 10&=e^(kt)\\&\textsf{Take natural logs:} \quad & \ln 10&=\ln e^(kt)\\&\textsf{Apply the power log rule:} \quad & \ln 10&=kt \ln e\\&\textsf{As $\ln e = 1$:} \quad & \ln 10&=kt\\&\textsf{Divide both sides by $k$:} \quad & t&=(1)/(k) \ln 10\end{aligned}`

Therefore, the equation for how long it will take for Veronica's money to increase by a factor of 10 in terms of t and k is:

`\boxed{t=(1)/(k)\ln 10}`