Question

a famous piece of artwork worth 17,000 increases in value 3.5% yearly. sonja uses v(t)=17000(3.5) to represent the value after t years, why is her equation incorrect and what is the correct answer?

Answer 1

Sonja's equation \( v(t) = 17000(3.5) \) is incorrect because it suggests that she is multiplying the initial value of the artwork by 3.5, which implies that the value would simply be a single calculation rather than compounding yearly.

The correct formula to represent the value of an asset that increases by a percentage each year is:

\[
v(t) = v_0 \cdot (1 + r)^t
\]

Where:
- \( v_0 \) is the initial value (in this case, $17,000)
- \( r \) is the annual growth rate (in this case, 3.5% or 0.035)
- \( t \) is the time in years

Therefore, the correct equation should be:

\[
v(t) = 17000 \cdot (1 + 0.035)^t
\]

Which simplifies to:

\[
v(t) = 17000 \cdot (1.035)^t
\]

This equation correctly accounts for the yearly compounding increase in the value of the artwork.