Chapter 4.1, Problem 61PPS

(a)

To determine

To Calculate and Graph:

The equation for the number of cell phone customers in billions, C , t years since 2007.

Graph this equation

Answer to Problem 61PPS

Linear equation for the number of cell phone customers in billions, C , t years since 2007. $C=0.525t+3.25$

Explanation of Solution

Given Information:

In 2007, 3.25 billion customers worldwide used a cell phone, and 1000 new customers signed up each minute.

New customers signed up in 1 minute = 1000

New customers signed up in 1 hour (60 minutes) = $1000\cdot 60$

New customers signed up in 1 day (24 hours) = $1000\cdot 60\cdot 24$

New customers signed up in 1 year (365 days) = $\begin{array}{l}1000\cdot 60\cdot 24\cdot 365\\ =525,600,000\\ =0.525billion \end{array}$

Let

C = Number of cell phone customers in billion

t = Years since 2007

m = Number of customers signed up each year in billion = 0.525

Formula Used:

Equation of slope-intercept form of a line is given by:

  $y=mx+b\mathrm{............................}(1)$

Where

  $\begin{array}{l}b=y-\mathrm{int}ercept\\ m=\frac{rise}{run}=slope\end{array}$

Calculation:

First , we form the equation :

Let

C = Number of cell phone customers in billion

t = Years since 2007

m = Number of customers signed up each year in billion = 0.525

  $\begin{array}{l}when\\ t=\mathrm{0...........}[that\_is\_year"2007"]\\ C=3.25billion\\ \Rightarrow C-intercept=3.25\end{array}$

Now, we form the equation :

Number of customers signed up each year in billion = 0.525

So, Number of customers signed up in t years since 2007 = 0. 525t

So, the linear equation ( slope-intercept form) for the number of cell phone customers in billions, C , t years since 2007:

  $\begin{array}{l}C=0.525t+b\\ C-intercept=3.25\\ \Rightarrow C=0.525t+\mathrm{3.25.....................}(2)\end{array}$

Linear equation for the number of cell phone customers in billions, C , t years since 2007. $\begin{array}{l}C=0.525t+3.25\\ or\\ C=0.5t+3.25\end{array}$

Graph:

To plot the graph for the equation

  $\begin{array}{l}C=0.5t+3.25\\ or\\ C=\frac{1}{2}t+\mathrm{3.25...........................}(2)\\ m=slope=\frac{rise}{run}=\frac{1}{2}\\ C-\mathrm{int}ercept=3.25\end{array}$

For the graph , we have point (0,3.25). First plot the point (0,3.25). From the point , rise 1 unit and then run to the right 2 units , since the slope is $\frac{1}{2}$ . Plot the point (2,4.25). Now join the two points to get the line of the equation (2).

  

(b)

To determine

To Calculate :

Number of cell phone customers in 2011.

Answer to Problem 61PPS

Number of cell phone customers in 2011 = 5.35 billion

Explanation of Solution

Given Information:

From part (a)

Linear equation for the number of cell phone customers in billions, C , t years since 2007. $C=0.525t+\mathrm{3.25........................}(2)$

Calculation:

Number of years from 2007 to 2011 = 2011-2007 = 4 years

Substituting t = 4 in equation (2)

  $\begin{array}{l}C=0.525t+\mathrm{3.25........................}(2)\\ \Rightarrow C=0.525(4)+\mathrm{3.25................}[t=4]\\ \Rightarrow C=2.1+\mathrm{3.25.........................}[simplify]\\ \Rightarrow C=5.35\end{array}$

Hence, the number of cell phone customers in 2011 = 5.35 billion

(c)

To determine

To Calculate : In which year , the number of customers will be 10 billion.

Answer to Problem 61PPS

10 billion customers will be in 2020.

Explanation of Solution

Given Information:

From part (a) Linear equation for the number of cell phone customers in billions, C , t years since 2007. $C=0.525t+3.25$

Calculate:

To the year in which 10 billion customers will be there:

  $\begin{array}{l}C=0.525t+\mathrm{3.25...............................}(2)\\ \Rightarrow 10=0.525t+\mathrm{3.25.........................}[C=10]\\ \Rightarrow 10=0.525t+\mathrm{3.25.........................}[simplify]\\ \Rightarrow 10-3.25=0.525t+3.25-\mathrm{3.25....}[subtract"3.25"each\_side]\\ \Rightarrow 6.75=0.525t\mathrm{...............................}[simplify]\\ \Rightarrow \frac{6.75}{0.525}=\frac{0.525}{0.525}t\mathrm{.............................}[divide"0.525"each\_side]\\ \Rightarrow 12.8571=t\mathrm{....................................}[simplify]\\ \Rightarrow t\approx 13\end{array}$

Hence, number of years since 2007 when there will be 10 billion customers = 13.

So, year = 2007+13 =2020.