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Between 2006 and 2016, the number of applications for patents, [tex]\( N \)[/tex], grew by about [tex]\( 3.5 \% \)[/tex] per year. That is, [tex]\( N'(t) = 0.035 N(t) \)[/tex].

a) Find the function that satisfies this equation. Assume that [tex]\( t = 0 \)[/tex] corresponds to 2006, when approximately 459,000 patent applications were received.

[tex]\[ N(t) = \square \][/tex]

b) Estimate the number of patent applications in 2021.

c) Estimate the rate of change in the number of patent applications in 2021.

Sagot :

GinnyAnswer
Sure, let's go through each part of the question step-by-step.

### Part a: Find the function that satisfies the differential equation [tex]\( N'(t) = 0.035 \, N(t) \)[/tex]

Given the differential equation [tex]\( N'(t) = 0.035 \, N(t) \)[/tex], we can recognize this as a first-order linear differential equation. The general solution to such an equation is of the form [tex]\( N(t) = N(0) e^{rt} \)[/tex], where [tex]\( r \)[/tex] is the growth rate (in this case 0.035), and [tex]\( N(0) \)[/tex] is the initial number of patent applications.

Since [tex]\( N(0) = 459000 \)[/tex] (the number of patent applications at [tex]\( t = 0 \)[/tex] or the year 2006), we can write the solution as:
[tex]\[ N(t) = 459000 \, e^{0.035t} \][/tex]

### Part b: Estimate the number of patent applications in 2021

To find the number of patent applications in 2021, we need to evaluate [tex]\( N(t) \)[/tex] at [tex]\( t = 2021 - 2006 = 15 \)[/tex] years after 2006.

Using the function we derived in part a:
[tex]\[ N(15) = 459000 \, e^{0.035 \times 15} \][/tex]

Evaluating this expression gives us the estimated number of patent applications in 2021:
[tex]\[ N(15) \approx 775920.61 \][/tex]

Thus, the estimated number of patent applications in 2021 is approximately 775,920.

### Part c: Estimate the rate of change in the number of patent applications in 2021

The rate of change in the number of patent applications is given by the derivative [tex]\( N'(t) \)[/tex]. From the differential equation, we know:
[tex]\[ N'(t) = 0.035 \, N(t) \][/tex]

To find the rate of change in 2021, we need to evaluate [tex]\( N'(t) \)[/tex] at [tex]\( t = 15 \)[/tex]:
[tex]\[ N'(15) = 0.035 \times N(15) \][/tex]

We already know [tex]\( N(15) \approx 775920.61 \)[/tex]:
[tex]\[ N'(15) \approx 0.035 \times 775920.61 \][/tex]
[tex]\[ N'(15) \approx 27157.22 \][/tex]

Thus, the estimated rate of change in the number of patent applications in 2021 is approximately 27,157 applications per year.

### Summary:
a) The function that satisfies the differential equation is:
[tex]\[ N(t) = 459000 \, e^{0.035t} \][/tex]

b) The estimated number of patent applications in 2021 is approximately:
[tex]\[ 775,920 \][/tex]

c) The estimated rate of change in the number of patent applications in 2021 is approximately:
[tex]\[ 27,157 \, \text{applications per year} \][/tex]
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