Answer:
16%
Step-by-step explanation:
To solve this we are using the standard growth equation:
[tex]y=a(1+b)^x[/tex]
Were
[tex]y[/tex] is the final value after [tex]x[/tex] years
[tex]a[/tex] is the initial value
[tex]b[/tex] is the growth factor (yearly rate of appreciation in our case) in decimal form
[tex]x[/tex] is the time in years
We know from our problem that gold coin appreciated in value from $200.00 to $475.00 in 6 years, so [tex]y=475[/tex], [tex]a=200[/tex], and [tex]x=6[/tex].
Let's replace the values in our equation and solve for [tex]b[/tex]:
[tex]y=a(1+b)^x[/tex]
[tex]475=200(1+b)^6[/tex]
[tex]\frac{475}{200} =(1+b)^6[/tex]
[tex]2.375=(1+b)^6[/tex]
[tex]\sqrt[6]{2.375} =\sqrt[6]{(1+b)^6}[/tex]
[tex]1+b=\sqrt[6]{2.375}[/tex]
[tex]b=\sqrt[6]{2.375}-1[/tex]
[tex]b=0.155[/tex]
which rounds to
[tex]b=0.16[/tex]
Since our appreciation rate is in decimal form, we need to multiply it by 100% to express it as percentage:
0.16*100% = 16%
We can conclude that the yearly appreciation rate of our gold coin is approximately 16%
