Question

no calculator is allowed on this question.
the value of a rare coin is found to double every 5 years. the initial value of the coin is $100.
identify the function that determines the value, v, of the coin after x years.
select one answer
a ( v(x) = 100left(\frac{2}{5}
ight)^x )
b ( v(x) = 100(2)^{x/5} )
c ( v(x) = 100(2)^{5x} )
d ( v(x) = (200)^{5x} )

Explanation:

Step1: Recall exponential growth formula

The general form of exponential growth is \( V(x) = a(b)^{t} \), where \( a \) is the initial value, \( b \) is the growth factor, and \( t \) is the time variable. Here, the initial value \( a = 100 \).

Step2: Determine the growth factor and time interval

The coin doubles every 5 years. So, when \( x = 5 \) years, the value should be \( 2\times100 = 200 \). We need to find the exponent for \( x \) years. Let's analyze the exponent: if the doubling time is 5 years, the number of doubling periods in \( x \) years is \( \frac{x}{5} \). So the growth factor \( b = 2 \), and the exponent is \( \frac{x}{5} \).

Step3: Form the function

Substituting \( a = 100 \), \( b = 2 \), and the exponent \( \frac{x}{5} \) into the exponential growth formula, we get \( V(x)=100(2)^{\frac{x}{5}} \).

Answer:

B. \( V(x) = 100(2)^{\frac{x}{5}} \)