Answer:
[tex]t = 9.05[/tex] weeks
Step-by-step explanation:
The mass of this particular substance can be modeled by the following exponential function:
[tex]m(t) = m(0)*e^{rt}[/tex]
In which [tex]m(t)[/tex] is the mass in function of time, [tex]m(0)[/tex] is the initial mass and r, in decimal, is the growth rate of the mass.
The problem states that:
The mass of a particular substance is known to grow exponentially at a rate of 17% per week. Its initial mass was 12 grams and, after t weeks, it weighed 56 grams. So:
[tex]r = 17% = 0.17[/tex]
[tex]m(0) = 12[/tex]
[tex]m(t) = 56[/tex]
We have to solve this equation for t. So:
[tex]m(t) = m(0)*e^{rt}[/tex]
[tex]56 = 12*e^{0.17t}[/tex]
[tex]e^{0.17t} = \frac{56}{12}[/tex]
[tex]e^{0.17t} = 4.67[/tex]
To solve for [tex]t[/tex], we put ln in both sides
[tex]ln e^{0.17t} = ln 4.67[/tex]
[tex]0.17t = 1.54[/tex]
[tex]t = \frac{1.54}{0.17}[/tex]
[tex]t = 9.05[/tex] weeks