Juan purchased an antique that had a value of $200 at the time of purchase. Each year, the value of the antique is estimated to increase 10% percent over its value the previous year. The estimated value of the antique, in dollars, 2 years after purchase can be represented by the expression 200a, where a is a constant. What is the value of a?

In order to obtain the value of a, you have to use The Exponential Growth Formula, because it represents a quantity increasing by a rate over a period of time, which is the given case.

The formula is:

[tex]Y=q(1+r)^{t}[/tex]

Y: The amount of the quantity after the period of time

q:The initial quantity

r: The rate or percentage of growth

t: The period of time

In this problem:

q=200 dollars

r=10%=0.1 (As a decimal number)

t=2 years

Replacing the values of q,r and t:

[tex]Y=200(1+0.1)^{2}[/tex]

But the value of Y was given as Y=200a, therefore:

[tex]200a=200(1.1)^{2}[/tex]

Dividing by 200 both sides:

a=(1.1)²

a=1.21