Question

A rumor spreads through a group of teenagers

according to the exponential model
N = 2 (1.81).7t
where N is the number of teenagers who have heard
the rumor t hours after it is first started.
(a) Find the number of teenagers who started the
rumor.
(b) Write down the number of teenagers who have
heard the rumor five hours after it is first started.
(C) Determine the length of time it would take for 150
teenagers to have heard the rumor. Give your answer
correct to the nearest minute.

Answer 1

a). N = 2

b). N = 16

c). t = 624 minutes

Explanation:

Exponential model that shows the rumor spread through a group is

`N=2(1.81)^(0.7t)`

Where N = Number of teenagers who have heard the rumors

t = Duration after the rumor first started

(a). At the time t = 0

`N=2(1.81)^((0.7* 0))`

N = 2

Therefore, 2 teenagers started the rumor.

(b). At t = 5 hours

N =
`2(1.81)^(0.7* 5)`

N = 15.96

N ≈ 16

16 teenagers heard the rumors after 5 hours.

(c). If N = 150

150 =
`2(1.81)^(0.7t)`

75 =
`(1.81)^(0.7t)`

log(75) = 0.7t(log1.81)

1.87506 = 0.7t(0.25767)

t =
`(1.87506)/(0.180369)`

t = 10.395 hours

t ≈ 624 minutes

Therefore, after 624 minutes or rumor will be heard by 150 teenagers.