Question

the graph shows calculations for potassium-argon dating. potassium-argon dating graph: y-axis remaining portion of potassium (1/1, 1/2, 1/4, 1/8, 1/16), x-axis billions of years (1.3, 2.6, 3.9, 5.2); text potassium (k) decays into argon (ar). which statement explains what geologists can learn from the graph? ○ they can estimate that all the potassium will have decayed within 6.5 billion years. ○ they can tell that 1/8 of the argon gas will be left after 3.9 billion years. ○ they can determine that a rock is 1.3 billion years old if 1/2 of the potassium has decayed. ○ they can see that potassium’s half-life varies from every 1.3 billion years to every 5.2 billion years.

Explanation:

1. Analyze Option 1: The graph shows potassium decays over time, but it doesn't indicate all potassium will decay within 6.5 billion years (the curve approaches zero but doesn't necessarily reach it fully in that time, and the logic is flawed as decay is exponential and never fully "all" in a finite time in theory for first - order decay like this).
2. Analyze Option 2: The graph shows the remaining portion of potassium. Argon is the decay product. After 3.9 billion years, the remaining potassium is 1/8, so the argon would be 7/8, not 1/8 left. So this is incorrect.
3. Analyze Option 3: The half - life of potassium - argon is about 1.3 billion years (since at 1.3 billion years, the remaining potassium is 1/2, meaning half has decayed). So if half the potassium has decayed, the time elapsed (age of the rock) is one half - life, which is 1.3 billion years. This matches the graph.
4. Analyze Option 4: The half - life of a radioactive isotope is constant. The graph shows the decay over multiple half - lives (1.3, 2.6, 3.9, 5.2 billion years correspond to 1, 2, 3, 4 half - lives respectively, since each 1.3 billion years half the potassium decays). So the half - life is constant at 1.3 billion years, not varying.

Answer:

They can determine that a rock is 1.3 billion years old if 1/2 of the potassium has decayed.