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The number of applications for​ patents, N, grew dramatically in recent​ years, with growth averaging about 4.6​% per year. That​ is, Upper N prime (t )equals0.046Upper N (t ). ​a) Find the function that satisfies this equation. Assume that tequals0 corresponds to 1980​, when approximately 117 comma 000 patent applications were received. ​b) Estimate the number of patent applications in 2025. ​c) Estimate the doubling time for Upper N (t ).

Respuesta :

Answer:

(a)[tex]N(t)=117000(1.046)^t[/tex]

(b)885,343

(c)15 years

Step-by-step explanation:

Given that the number of applications for​ patents, N, grew dramatically in recent​years, with growth averaging about 4.6​% per year.

Part A

The function which satisfies the equation given that that t = 0 corresponds to 1980​, when approximately 117,000 patent applications were received is given by:

[tex]N(t)=117000(1+0.046)^t\\N(t)=117000(1.046)^t[/tex]

where:

  • N(t) is the number of patent applications received at any particular year,
  • t is the number of years after 1980.

Part B

In 2025, there are 2025 - 1980 = 45 years after 1980.

The number of patent applications 45 years after 1980 is given by:

[tex]N(t)=117000(1.046)^t\\N(45)=117000(1.046)^{45}\\\approx 885343[/tex]

Part C

The doubling time for N(t) is the time it takes the number of patents to be

2 X 117,000 = 234,000

When N(t)=234000

[tex]234000=117000(1.046)^t\\1.046^t=\frac{234000}{117000} =2\\$Changing to Logarithm\\log _{1.046}2=t\\\frac{Log 2}{Log 1.046}=t\\ t=15.41\approx 15 years[/tex]

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