Question

The rate at which ants arrive at a picnic is modeled by the function A, where A(t) is measured in ants per minute and t is measured in minutes. How many ants arrive at the picnic during the time interval 0≤t≤30?

a) 600 ants
b) 900 ants
c) 1,200 ants
d) 1,800 ants

Answer 1

The total number of ants arriving at a picnic over 30 minutes is calculated by integrating the rate function A(t) from 0 to 30 minutes. Without the explicit function, we cannot perform this calculation. The concept is analogous to integrating an arrival rate over time to get the total number of arrivals.

Step-by-step explanation:

The question pertains to determining the total number of ants that arrive at a picnic over a specified time interval using a given rate function, A(t). This involves calculating an integral over the time period from t = 0 to t = 30 minutes. The integral of the rate function A(t) gives the total quantity, in this case, the total number of ants that arrive at the picnic within that time frame. To provide an answer, the specific form of the function A(t) is required. Without the explicit rate function, we cannot integrate to find the total number of ants.

However, we can consider the provided information about arrival rates. For instance, if one customer arrives every two minutes on average, then we expect 15 customers in 30 minutes. But customers are not ants, and the arrival rate can differ. This analogy conveys the general principle that to determine the total quantity of arrivals, one must integrate the rate function over the given period. Simply put, the number of arrivals is the area under the rate function curve between two points in time.