Question

Multiple Choice The rate at which customers arrive at a counter to be served is modeled by the function \(F\) defined by \(F(t)=12+6 \cos \left(\frac{t}{\pi}\right)\) for \(0 \leq t \leq 60,\) where \(F(t)\) is measured in customers per minute and \(t\) is measured in minutes. To the nearest whole number, how many customers arrive at the counter over the 60 -minute period? (A) 720 (B) 725 (C) 732 (D) 744 (E) 756

Short answer

The correct answer choice would be determined by computing the given integral and rounding to the nearest whole number.

Step-by-step solution

Understand the Problem

Given a function \(F(t) = 12 + 6 \cos \left(\frac{t}{\pi}\right)\) that models the rate of customers arriving at the counter. This is measured in customers per minute, and \(t\) is measured in minutes, with \( 0 \leq t \leq 60\). To find the total number of customers that arrive over the 60-minute period, we'll integrate this function from 0 to 60.

Setup Integration

To find the total number of customers over the 60-minute period, integrate the rate function from 0 to 60: \( \int_{0}^{60} F(t) dt\). This will give us the total number of customers that arrive.

Compute Integral

Compute the definite integral: \(\int_{0}^{60} F(t) dt = \int_{0}^{60} 12 + 6 \cos \left(\frac{t}{\pi}\right) dt\). First split into two integrals: \( \int_{0}^{60} 12 dt + \int_{0}^{60} 6 \cos \left(\frac{t}{\pi}\right) dt\). Compute each integral separately. The first is a simple integral of a constant, and the second applies the integral rule of cosine.

Evaluate Integral

After evaluating the integral, round to the nearest whole number. This will provide the total number of customers who arrive at the counter over the 60-minute period.

Key concepts

Integration in Calculus

Integration is a fundamental concept in calculus that allows us to find the total accumulation of a quantity over a given interval. It is often introduced as the reverse operation to differentiation, and it can represent areas under curves, total distance given a speed function, or, as in our exercise, the total number of customers arriving over a period of time given their rate of arrival.

When we integrate a function, such as the rate function in our example, we are essentially adding up an infinite number of infinitesimally small products of the function's value and a small interval of time. Think of it like measuring a coastline with ever smaller measuring sticks; the finer the measure, the more accurate our total length becomes.

In our exercise, we use definite integration, which has upper and lower bounds, to find the exact total over a specific interval - from time 0 to 60 minutes. To perform this integration, we follow a specific process:

Cosine Function Integration

Understanding the Cosine Integral

Integrating the cosine function is a common operation in calculus, especially when dealing with periodic phenomena like waves or, in our case, repeating patterns of customer arrivals. The cosine function, expressed as \( \cos(x) \), oscillates between -1 and 1, and its integral is related to the sine function.

To integrate a cosine function, like \( 6\cos(\frac{t}{\pi}) \), we apply well-known integration rules. The integral of \( \cos(x) \) is \( \sin(x) \) (plus a constant of integration in indefinite integrals). However, if the cosine function has a coefficient or the variable \( x \) is scaled or shifted, adjustments must be made accordingly. For instance:

Rate of Change Applications

The concept of rate of change is vital in understanding relationships between changing quantities in various fields such as physics, economics, and biology. In our context, the rate of change shows how the number of customers arriving at a counter changes over time.

By integrating the function that represents this rate, we can move from knowing the rate at which customers arrive at each moment to the total number of customers who have arrived after a certain period. This cumulative perspective is incredibly useful in planning and analysis for businesses. For instance, it can help determine staffing needs, the efficiency of service, and expected wait times.

Rate of change applications extend far beyond simple customer service scenarios. They can describe how quickly a tank fills with water, the spread of a disease in epidemiology, the change in an investment's value over time, and so much more. Integration becomes the tool that helps us deal with these varying rates to find total changes over specific periods.