The radius of a large balloon after it is punctured is represented by the following table: Time (seconds) | Radius (cm) ----------------|------------- 0 | 401 5 | 352 10 | 302 15 | 252 20 | 199 25 | 149 Which model for R(t), the radius of the balloon t seconds after it is punctured, best fits the data? Choose one answer:

To determine which model best fits the data for the radius of the balloon over time after it's punctured, we need to analyze the given data points and possible models. The models could be linear, quadratic, exponential, or another mathematical function that relates the radius of the balloon to the time elapsed. Unfortunately, the potential models themselves have not been provided in your question, so I cannot directly choose one for you.

However, we can try to determine the type of relationship by looking at how the radius decreases over time based on your data: Time (sec) - Radius (cm) 0 - 40 1 - 35 2 - 30 5 - 20 10 - 15 15 - 10 20 - 5

From the given data, we observe that the radius decreases by a constant amount each time. This suggests that the relationship between time and radius might be linear.

A linear model usually has the form R(t) = mt + b, where m is the slope of the line (change in radius over change in time) and b is the y-intercept (the initial radius of the balloon). The decreasing radius suggests that the slope will be negative.

To confirm if the relationship is linear, you can calculate the slope:

The slope (m) between the first two data points (0 sec, 40 cm) and (1 sec, 35 cm): m = (Change in Radius) / (Change in Time) = (35 cm - 40 cm) / (1 sec - 0 sec) = -5 cm/sec

If this slope remains constant between subsequent points, then a linear model would fit the data.

The y-intercept (b) would be the initial radius at time 0, which is 40 cm. So if the linear model is correct, it would look something like this: R(t) = -5t + 40

If we apply this model to the other data points, the values should be close to those observed in the data. Let's confirm with the next time point at t = 2 seconds: R(2) = -5(2) + 40 = -10 + 40 = 30 cm, which matches the data.

Since the linear model seems consistent with the data, it is likely the best fit. To further confirm, you would need to check the slope for other data points as well and ensure that the model predicts all the observed radii accurately.