Answer: 36 years
Step-by-step explanation:
Exponential equation to represent growth:-
[tex]y=A(1+r)^t[/tex] , where A is the initial value , r is the rate of growth and t is the time period.
Given : A rare coin appreciates at a rate of 5.2% a year. If the initial value of the coin is $500.
i.e. Put A= 500 and r= 0.052 in the above formula.
The amount after t years:
[tex]y=500(1.052)^t[/tex]
Inequality for value cross $3,000 mark:
[tex]3000<500(1.052)^t[/tex]
Divide both sides by 500
[tex]6<(1.052)^t[/tex]
Taking log on both sides , we get
[tex]\log6<t\ \log(1.052)\\\\=0.778151250384< t(0.0220157398177)\\\\ t>\dfrac{0.778151250384}{0.0220157398177}=35.345223773\\\\t\approx36[/tex]
Hence, it will take approx 36 years to cross the $3,000 mark.
