Question

would (0,14) be the y-intercept point?
yes! (0, 14) is correct for the y-intercept since you start with 14 ounces at time t = 0. now youll need at least one more point to graph the exponential curve. what information from the problem could help you find another point?
(1,24)
think about what happens to caffeine over time - does the amount in your body increase or decrease? also, remember the problem mentions a half-life of 6 hours. what does that tell you about the caffeine amount at t = 6?
ask the ai coach a question

Explanation:

The problem involves analyzing caffeine decay (since half - life is mentioned, so it's a decay process, not growth as (1,24) would imply growth which is incorrect for caffeine). The half - life of 6 hours means that at \(t = 6\), the amount of caffeine should be half of the initial amount. The initial amount at \(t = 0\) is 14 ounces, so at \(t=6\), the amount should be \(14\times\frac{1}{2}=7\) ounces. So the point \((6,7)\) can be found using the half - life information. Also, caffeine in the body decreases over time (it metabolizes), so the amount should decrease as time \(t\) increases, so \((1,24)\) is incorrect as it shows an increase. To find another point, we can use the half - life: at \(t = 6\) (half - life time), the caffeine amount is half of the initial. So the information about the half - life (6 hours) and the fact that caffeine amount decreases over time helps. We can calculate the amount at \(t = 6\) as \(14\times\frac{1}{2}=7\), so the point \((6,7)\) is another point.

Answer:

To find another point, we use the half - life (6 hours) of caffeine. Since caffeine decays, at \(t = 6\) (half - life), the amount is half of the initial (\(14\) ounces). So the amount at \(t = 6\) is \(14\times\frac{1}{2}=7\) ounces, giving the point \((6,7)\). Also, caffeine amount decreases over time (so \((1,24)\) is incorrect as it implies increase), and the half - life info helps calculate the amount at \(t = 6\).