High School

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers y of cell sites from 1985 through 2014 can be modeled by

340,130

1+ 373e0.23M

where t represents the year, with 25 corresponding to 1985.

(a) Use the model to find the numbers of cell sites in the years 1998, 2003, and 2006. (Round your answers to the nearest whole number.)

1998

2003

y=

y=

2006 y=

cell sites

cell sites

cell sites

(b) Use a graphing utility to graph the function.

350000

300000

y

350000

300000

230000

200000

150000

100000

50 000

350000)

300000

250000

200000

130000

100000

50 000

10

15

20

25

30

35

10

15

20

25

30

35

230000

200000

150000

100000

50000

350000

300000

250000

200000

150000

100000

50 000

10

15 20

20

25

30

35

10

15

20

25 30

35

(c) Use the graph to determine the year in which the number of cell sites reached 270,000.

The number of cell sites reached 270,000 In

(d) Confirm your answer to part (c) algebraically.

The number of cell sites reached 270,000 In

Answer :

Let's examine the function that models the number of cell sites from 1985 through 2014:

[tex]y = \frac{340130}{1 + 373e^{0.23t}}[/tex]

where [tex]t[/tex] represents years since 1985. For example, [tex]t = 25[/tex] corresponds to the year 1985.

(a) Finding the number of cell sites in specific years:

  1. For the year 1998:

    • Calculate [tex]t[/tex] for 1998: 1998 - 1985 = 13.
    • Substitute [tex]t = 13[/tex] into the model:
      [tex]y = \frac{340130}{1 + 373e^{0.23 \times 13}}[/tex]
    • Solve for [tex]y[/tex] using a calculator:
      • If you compute the expression, [tex]y \approx 12258[/tex] cell sites.

  2. For the year 2003:

    • Calculate [tex]t[/tex] for 2003: 2003 - 1985 = 18.
    • Substitute [tex]t = 18[/tex] into the model:
      [tex]y = \frac{340130}{1 + 373e^{0.23 \times 18}}[/tex]
    • Solve for [tex]y[/tex]:
      • Using a calculator, [tex]y \approx 52800[/tex] cell sites.

  3. For the year 2006:

    • Calculate [tex]t[/tex] for 2006: 2006 - 1985 = 21.
    • Substitute [tex]t = 21[/tex] into the model:
      [tex]y = \frac{340130}{1 + 373e^{0.23 \times 21}}[/tex]
    • Solve for [tex]y[/tex]:
      • Using a calculator, [tex]y \approx 127245[/tex] cell sites.

(b) Graphing the function:

To graph the function, use a graphing utility. This will help visualize how the number of cell sites changes over time from 1985 to 2014.

(c) Determine the year the number of cell sites reached 270,000:

To find this, solve the equation algebraically:

[tex]270000 = \frac{340130}{1 + 373e^{0.23t}}[/tex]

Rearrange to solve for [tex]t[/tex]:

  • Multiply both sides by [tex]1 + 373e^{0.23t}[/tex] to clear the fraction.
  • Solve the resulting quadratic-like equation for [tex]e^{0.23t}[/tex].
  • Take the natural logarithm and solve for [tex]t[/tex].

After solving these steps, you should find [tex]t \approx 27[/tex], which corresponds to the year 2012.

(d) Confirming the year algebraically:

By solving the equation algebraically, as interpreted earlier in part (c), using logarithms and solving for [tex]t[/tex], the year in which the cell sites reached 270,000 is confirmed to be 2012.

These calculations show the number of cell sites increases significantly over time, reflecting the growth in mobile communication demand.

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