Question

the population of a country is growing according to the function P=1,500,000e^(0.07t). where t is the number of the year since 1990. Determine the inverse function and the year in which the population will reach 3,730,000

Answer 1

Answer:

$t=\frac{1}{0.07}\ln \frac{P}{1500000}$

$t\approx 13\ years$

Step-by-step explanation:

Inverse function: Function in which dependent variable becomes independent vice versa, is known as inverse function.

$P=1500000e^{0.07t}\\\\Take\ \ln\ both\ side\\\\\ln P=\ln 1500000e^{0.07t}\\\\\ln P=\ln 1500000+\ln e^{0.07t}\ \ \ \ \ \ \ (as\ \ln ab=\ln a+\ln b)\\\\\ln P=\ln 1500000+0.07t\ \ \ \ \ \ \ \ \ \ \ \ \ (as \ln e^a=a)\\\\0.07t=\ln P-\ln 1500000\\\\0.07t=\ln \frac{P}{1500000}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (as\ln \frac{a}{b}=\ln a-\ln b)\\\\t=\frac{1}{0.07}\ln \frac{P}{1500000}\\\\Here\ P=3730000\\\\t=\frac{1}{0.07}\ln \frac{3730000}{1500000}\\\\t=\frac{1}{0.07}\times 0.910943=13.01$

$t\approx 13\ years$