Answer: 36 years
Step-by-step explanation:
Exponential equation to represent growth:-
, 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:
![y=500(1.052)^t](https://tex.z-dn.net/?f=y%3D500%281.052%29%5Et)
Inequality for value cross $3,000 mark:
![3000](https://tex.z-dn.net/?f=3000%3C500%281.052%29%5Et)
Divide both sides by 500
![6](https://tex.z-dn.net/?f=6%3C%281.052%29%5Et)
Taking log on both sides , we get
![\log6\dfrac{0.778151250384}{0.0220157398177}=35.345223773\\\\t\approx36](https://tex.z-dn.net/?f=%5Clog6%3Ct%5C%20%5Clog%281.052%29%5C%5C%5C%5C%3D0.778151250384%3C%20t%280.0220157398177%29%5C%5C%5C%5C%20t%3E%5Cdfrac%7B0.778151250384%7D%7B0.0220157398177%7D%3D35.345223773%5C%5C%5C%5Ct%5Capprox36)
Hence, it will take approx 36 years to cross the $3,000 mark.