between 2006 and 2016, the number of applications for patents, n, grew by about 3.9% per year. that is…
between 2006 and 2016, the number of applications for patents, n, grew by about 3.9% per year. that is, n(t)=0.039n(t). a) find the function that satisfies this equation. assume that t = 0 corresponds to 2006, when approximately 463,000 patent applications were received. b) estimate the number of patent applications in 2022. c) estimate the rate of change in the number of patent applications in 2022.
Answer
Explanation:
Step1: Solve the differential equation
The differential equation $N^{\prime}(t)=0.039N(t)$ is a first - order separable differential equation of the form $\frac{dN}{dt}=kN$, where $k = 0.039$. The general solution of $\frac{dN}{dt}=kN$ is $N(t)=N_0e^{kt}$, where $N_0$ is the initial value. Given $t = 0$ corresponds to 2006 and $N(0)=N_0 = 463000$, so the function is $N(t)=463000e^{0.039t}$.
Step2: Calculate $t$ for 2022
Since $t = 0$ corresponds to 2006, for 2022, $t=2022 - 2006=16$.
Step3: Estimate the number of patent applications in 2022
Substitute $t = 16$ into $N(t)$: $N(16)=463000e^{0.039\times16}=463000e^{0.624}$. Using a calculator, $e^{0.624}\approx1.866$, so $N(16)=463000\times1.866 = 863958$.
Step4: Find the derivative of $N(t)$
Since $N(t)=463000e^{0.039t}$, then $N^{\prime}(t)=463000\times0.039e^{0.039t}=18057e^{0.039t}$.
Step5: Estimate the rate of change in 2022
Substitute $t = 16$ into $N^{\prime}(t)$: $N^{\prime}(16)=18057e^{0.039\times16}=18057e^{0.624}$. Since $e^{0.624}\approx1.866$, $N^{\prime}(16)=18057\times1.866 = 33694.362$.
Answer:
a) $N(t)=463000e^{0.039t}$ b) 863958 c) 33694.362