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. a) n(t)=463000e^0.039t b) the number of patent applications in 2022 will be 864,133 (round to the nearest whole number as needed.) c) the rate of change in the number of patent applications in 2022 is about (round to the nearest whole number as needed.) year(s) per application. application(s) per year.
Answer
Explanation:
Step1: Recall the formula for exponential - growth
The differential equation $N'(t)=kN(t)$ has the solution $N(t)=N_0e^{kt}$, where $N_0$ is the initial value and $k$ is the growth rate. Given $k = 0.039$ and $N_0=463000$ (when $t = 0$ in 2006), the function is $N(t)=463000e^{0.039t}$.
Step2: Calculate $t$ for 2022
The year 2022 corresponds to $t=2022 - 2006=16$. Substitute $t = 16$ into $N(t)$: $N(16)=463000e^{0.039\times16}$. $N(16)=463000e^{0.624}\approx463000\times1.86681\approx864133$.
Step3: Find the rate - of - change in 2022
Since $N'(t)=0.039N(t)$, substitute $t = 16$ (for 2022). First, we know $N(16)\approx864133$. Then $N'(16)=0.039\times N(16)$. $N'(16)=0.039\times864133\approx33601$.
Answer:
33601 applications per year