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

[tex]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\\\\[/tex]

[tex]t\approx 13\ years[/tex]

MM5iveee MM5iveee · Answer 2 · 2021-04-12T07:04:44+02:00

Answer: (B) The function is t=in(p/1,500,000)/0.07 , and the year is 2003 .

Step-by-step explanation: PLATO/EDMENTUM

To find the inverse, solve for the independent variable, t, in the initial function:

To determine the year in which the population will reach 3,730,000, first find the number of years, t, by substituting P = 3,730,000 in the inverse function:

Since t is the number of years since 1990, the year in which the population reached 3,730,000 is 1990 + 13 = 2003.