A rare coin appreciates at a rate of 5.2% a year. If the initial value of the coin is $500, after how many years will itsvalue cross the $3,000 mark? Show the formula that models the value of the coin after t years.
Accepted Solution
A:
Answer: 36 yearsStep-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.