Question

A rumor spreads among a group of 400 people. The number of people, N(t), who have heard the rumor by the time t in hours since the rumor started to spread can be approximated by a function of the form
N(t) = 400 / (1+399e^(-0.4t))
Approximately how long will it take until half the people have heard the rumor?

Answer 1

Answer:

  about 15 hours

Step-by-step explanation:

You want to find t such that N(t)=200. Fill in the equation with that information and solve for t.

  200 = 400/(1 +399e^(-0.4t))

  1 +399e^(-0.4t) = 400/200 = 2 . . . . . multiply by (1+399e^(-0.4t))/200

  399e^(-0.4t) = 1 . . . . . . . . . . . . . . . . . . subtract 1

  e^(-0.4t) = 1/399 . . . . . . . . . . . . . . . . . .divide by 399

  -0.4t = ln(1/399) . . . . . . . take the natural log

  t = ln(399)/0.4 ≈ 14.972 . . . . . . . divide by -0.4, simplify

Rounded to tenths, it will take 15.0 hours for half the people to have heard the rumor.