Question

Can someone please help with this question for math, thank you!

Answer 1

The continous exponential growth model has the form:

$P=P_0e^{rt}$

where Po is the starting population, P is the total population after time t, r is the rate growth, t is the time and e is Euler's number. In our case, r = 0.078, P is 2Po and we need to find the time t. By substituting these values, we get

$2P_0=P_0e^{0.078\text{ t}}$

By moving the initial population Po to the left hand side, we get

$\frac{2P_0}{P_0}=e^{0.078\text{ t}}$

so we can cancel out Po and get

$2=e^{0.078\text{ t}}$

Now, by applying natural logarithm in both sides, we have

$\ln 2=\ln e^{0.078\text{ t}}$

since natural logarithm is the inverse of the exponential function , we get

$\ln 2=0.078t$

then, by moving the coefficient of t to the left hand side,we obtain

$\frac{\ln 2}{0.078}=t$

since ln2 is 0.693m, we have

$t=\frac{0.693}{0.078}$

finally, the time is

$t=8.886\text{ hours}$

Then, by rounding to the nearest hundredth, the answer is 8.89 hours