Mathematics

Problem

An internet service provider charges a monthly fee

P(d)

(in dollars) based on a customer’s data usage

d

(in gigabytes), with

0<d\le 800

P(d)=\begin{cases} 30+0.10d, & 0<d\le 200,\\ 50+0.08d, & 200<d\le 400,\\ 100+0.05d+0.0001(d-400)^{2}, & 400<d\le 800. \end{cases}

Customers are interested in selecting a usage level that minimizes the average cost per gigabyte, defined by

A(d)=\dfrac{P(d)}{d}

. a. Part 1: Express

A(d)

explicitly for each of the three usage intervals provided in the definition of

P(d)

. b. Part 2: On every interval where

A

is differentiable, compute

\dfrac{dA}{dd}

and determine all critical values that fall within the respective open intervals. c. Part 3: Evaluate

A(d)

at each critical value obtained in part (b) and at the junction points

d=200,\,400,\,800

. d. Part 4: Using all candidate values from part (c), determine the data usage level that minimizes the average cost per gigabyte over

0<d\le 800

. State the minimizing usage and the corresponding minimum average cost, and discuss the practical implications for a customer’s data plan choice.