Question

a collector’s item is purchased for $150 and its value increases by 3% each year. which graph can be used to determine approximately how many years it will take for the value to double?

Explanation:

Step1: Define exponential growth formula

The value of the item follows exponential growth: $V(t) = V_0(1+r)^t$, where $V_0=150$, $r=0.03$, and $V(t)=300$ (double the initial value).

Step2: Set up equation for doubled value

$$300 = 150(1+0.03)^t$$

Step3: Simplify the equation

Divide both sides by 150: $2 = (1.03)^t$

Step4: Solve for t using logarithms

Take natural log of both sides: $\ln(2) = t\ln(1.03)$, so $t = \frac{\ln(2)}{\ln(1.03)}$

Step5: Calculate t

$t \approx \frac{0.6931}{0.0296} \approx 23.45$

Step6: Match to correct graph

The graph with the point $(23.45, 300)$ (middle graph) corresponds to this result, as exponential growth curves upward over longer time.

Answer:

It will take approximately 23.45 years for the value to double, and the correct graph is the middle one with the point $(23.45, 300)$.