Answer:
The correct answer is 54.76 years.
Step-by-step explanation:
The national health care expenditure (H) , in billion of dollars is modeled by
H = 29.57 × [tex]e^{0.0970t}[/tex].
To measure the time before which national health expenditure reach 6000 billion dollars.
Thus putting the value of H = 6000 in the above modeled equation we get,
⇒ 6000 = 29.57 × [tex]e^{0.0970t}[/tex]
⇒ [tex]\frac{6000}{29.57}[/tex] = [tex]e^{0.0970t}[/tex]
⇒ 202.908 = [tex]e^{0.0970t}[/tex]
Taking logarithm with the base of e (㏑) both sides we get,
⇒ ㏑ 202.908 = ㏑ [tex]e^{0.0970t}[/tex]
⇒ ㏑ 202.908 = 0.0970 × t
⇒ 0.0970 × t = 5.312
⇒ t = [tex]\frac{5.312}{0.0970}[/tex]
⇒ t = 54.76.
Thus the total time required before which national health expenditure reach 6000 billion dollars is 54.76 years.
