Question

The customer service department of a wireless cellular provider has found that on Wednesdays, the polynomial function \(C(t)=-0.0815 t^{4}+t^{3}+12 t\) approximates the number of calls received by any given time, where \(t\) represents the number of hours that have passed during the workday . Based on this function, how many calls can be expected by the end of one 10 -hour workday?

Short answer

305 calls can be expected by the end of a 10-hour workday.

Step-by-step solution

Understand the function

Identify the polynomial function given in the problem. The function provided is: \[C(t) = -0.0815 t^{4} + t^{3} + 12 t\] where \(t\) represents the number of hours.

Substitute the value of t

We need to find the number of calls expected by the end of a 10-hour workday. Substitute \( t = 10 \) into the polynomial function. \[C(10) = -0.0815(10)^{4} + (10)^{3} + 12(10)\]

Calculate each term individually

First, calculate \( 10^4 \): \[10^4 = 10000\] Then multiply by -0.0815: \[-0.0815 \times 10000 = -815\] Next, calculate \( 10^3 \): \[10^3 = 1000\] And multiply by 1: \[ 1 \times 1000 = 1000\] Finally, calculate 12 x 10: \[12 \times 10 = 120\]

Sum up the results

Sum the results of the individual calculations: \[-815 + 1000 + 120\]

Find the total number of calls

Add the numbers together: \[1000 - 815 + 120 = 305\]

Key concepts

substitution method in algebra

The substitution method in algebra is a powerful tool. It helps us find specific values of unknowns by replacing variables with known quantities. In our given problem, we were asked to find the number of calls the customer service department receives by the end of a 10-hour workday. The polynomial function provided is: \[C(t) = -0.0815 t^{4} + t^{3} + 12 t\] By substituting the value of \( t = 10 \) into this equation, we can compute the specific value for a 10-hour period. This substitution simplifies the problem, turning it into a straightforward arithmetic calculation. This method is essential for solving many kinds of algebra problems, especially when dealing with equations and functions.

calculating polynomial values

Calculating polynomial values might sound complex, but breaking it down into steps simplifies the process. Here, we needed to evaluate the function by replacing \( t \) with 10. The function is: \[C(10) = -0.0815(10)^{4} + (10)^{3} + 12(10)\] Here's how you do it step-by-step:

• First, calculate each individual term.
• Start with \(10^4\): \(10^4 = 10000\).
• Multiply by -0.0815: \(-0.0815 \times 10000 = -815\).
• Next, calculate \(10^3\): \(10^3 = 1000\).
• Multiply this by 1: \(1 \times 1000 = 1000\).
• Finally, calculate the term \(12 \times 10 = 120\).

After calculating all the terms individually, sum up these results to find the final value. So: \[-815 + 1000 + 120 = 305\] This means that by the end of a 10-hour workday, the customer service department can expect 305 calls.

mathematical modeling in customer service

Mathematical modeling is a crucial technique in various fields, including customer service. By developing a mathematical model, like the polynomial function in this exercise, we can predict and analyze real-world scenarios. In our case, the function: \[C(t) = -0.0815 t^{4} + t^{3} + 12 t\] helps us understand how calls vary over the hours in a workday. This polynomial model considers different rates of change over time. Using such models, companies can plan resources and manage staffing needs effectively. Predicting call volumes ensures better preparedness, leading to improved customer service and satisfaction. These predictive analytics tools, based on mathematical models, form the backbone of efficient operational strategies in many industries.