The logistic equation has been used to model the spread of technology. let n∗ be the number of ranchers in uruguay, and n(t) the number who have adopted a new pasture technology. the rate of adoption dn/dt is proportional to the number who have adopted the technology, and the fraction who have not (and thus are susceptible to changing). so the equation is dn dt = αn 1 − n n∗ according to a study by banks (1993), n∗ = 17, 015, n(0) = 141, α = 0.49 per year. determine how long it takes for the new technology to be adopted by 80% of the population of ranchers.

12.6 years

Step-by-step explanation:

The equation of adoption of the pastures is

Rearranging we have

Integrating

At t=0, N(0)=141

The 80% of the population of ranchers represents 0.8*17015=13612 ranchers, so the time needed to reach that ammount of adoption is

The time it takes for a 80% adoption is 12.6 years

answer: ok

step-by-step explanation:

(a) {a₁, a₁r, a₁r², … a₁rⁿ⁻¹, …}

(b) $59 398

step-by-step explanation:

(a) the annual sequence of profits

aₙ = a₁rⁿ⁻¹

the model for the sequence of profits is {a₁, a₁r, a₁r², … a₁rⁿ⁻¹, …} .

(b) profit after 15 yr

  a₁ = $ 30 000

  r = 1.05

a₁₅ = 30 000 × 1.05¹⁴

a₁₅ = 30 000 × 1.979 93

a₁₅ = $59 398

the profit in year 15 will be $59 398.