Question

The net worthw(t) of a company is growing at a rate of′(t) = 2000−121t2dollarsper year, wheretis in years since 1990.(a) If the company is worth $40,000 in 1990, how much is it worth in 2000

Answer 1

Answer:

The worth of the company in 2000 is $56,000.

Step-by-step explanation:

The growth rate of the company is:

$f'(t)=2000-12t^{2}$

To determine the worth of the company in 2000, first compute the change in the net worth during the period 1990 (t = 0) to 2000 (t = 10) as follows:

$\int\limits^{10}_{0} {2000-12t^{2}} \, dt =\int\limits^{10}_{0} {2000} \, dt-12\int\limits^{10}_{0} {t^{2}} \, dt=2000 |t|^{10}_{0}-12|\frac{t^{3}}{3}|^{10}_{0}\\=(2000\times10)-(4\times10^{3})\\=20000-4000\\=16000$

The increase in the company's net worth from 1990 to 2000 is $16,000.

If the company's worth was $40,000 in 1990 then the worth of the company in 2000 is:

Worth in 2000 = Worth in 1990 + Net increase in company's worth

                        $=40000+16000\\=56000$

Thus, the worth of the company in 2000 is $56,000.