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?
(graphs are shown with different points and lines, including (-23.45, 150), (23.45, 300), (1.13, 300) on respective graphs)

Explanation:

Step1: Identify the growth model

The value of the collector's item increases by 3% each year, so it follows an exponential growth model. The formula for exponential growth is \( y = a(1 + r)^x \), where \( a = 150 \) (initial value), \( r = 0.03 \) (growth rate), so the equation is \( y = 150(1.03)^x \). We want to find when \( y = 300 \) (double the initial value of 150).

Step2: Analyze the graphs

• The first graph has a negative x - value, which doesn't make sense for time (years can't be negative in this context as we start from the purchase time).
• The second graph: Let's check the equation. If we set \( y = 150(1.03)^x = 300 \), we can solve for \( x \) by dividing both sides by 150: \( (1.03)^x = 2 \). Taking the natural logarithm of both sides: \( x=\frac{\ln(2)}{\ln(1.03)}\approx\frac{0.6931}{0.0296}\approx23.45 \). So when \( y = 300 \), \( x\approx23.45 \), which matches the point \( (23.45, 300) \) on the second graph. The graph is an exponential growth curve (since it's increasing and the rate is percentage - based, so exponential, but the graph here looks linear? Wait, no, maybe it's a linear approximation or the way the graph is plotted, but the key is the x - value for \( y = 300 \) is around 23.45, which is positive and makes sense for time (years after purchase).
• The third graph has a very small x - value (1.13) which is inconsistent with the exponential growth calculation (we know from the rule of 72, 72/3≈24, so around 23 - 24 years, not 1.13).

Answer:

The second graph (with the point (23.45, 300))