The net worth, $f(t)$, of a company is growing at a rate of $f^{\prime}(t)=2000-12 t^{2}$ dollars per year, where $t$ is in years since 2020 . How is the net worth of the company expected to change between 2020 and 2030 ? If the company is worth $\$ 40,000$ in 2020 , what is it worth in 2030?
Change in net worth of the company $=\$$
If the company is worth $\$ 40,000$ in 2020 , then the net worth of the company in 2030 is $\$$
Answer
Final Answer: The change in net worth of the company between 2020 and 2030 is \(\boxed{\$16,000}\). The net worth of the company in 2030 is \(\boxed{\$56,000}\).
Step 1 ：The problem is asking for two things. First, it wants to know the change in the net worth of the company between 2020 and 2030. This can be calculated by integrating the rate of change function, \(f^{\prime}(t)=2000-12 t^{2}\), from 0 to 10 (representing the years from 2020 to 2030).
Step 2 ：Second, it wants to know the net worth of the company in 2030. This can be calculated by adding the change in net worth to the initial net worth of the company in 2020, which is given as $40,000.
Step 3 ：By calculating, the change in net worth of the company between 2020 and 2030 is $16,000.
Step 4 ：By adding the change in net worth to the initial net worth of the company in 2020, the net worth of the company in 2030 is $56,000.
Step 5 ：Final Answer: The change in net worth of the company between 2020 and 2030 is \(\boxed{\$16,000}\). The net worth of the company in 2030 is \(\boxed{\$56,000}\).
