I have the answer. I just need the explanation?

Question

In 1980, about 27,7000 cases of TB were reported in the US. In 997, there were 19,9000cases.

Write an exponential decay function that models the number of reported cases of TB as a function of time.

Answer: (p0) = 27,7000(0.9807)^t

How was the 0.9807 calculated?

Please explain. I have been racking my brains for two days.

Thank you.

Answer 1

You do it with logarithms and an exponential, or with fractional exponents, whichever you find easier.

(1) log(27700) = 10.2292, log(19900) = 9.8985, change = -0.3307, divide by 17 = -0.01945, exp(-0.01945) = 0.9807.

(2) C1 = 27700, C2 = 19900 = C1 * K^17, therefore K^17 = 19900/27700 = 0.7184. Take the 17th root of each side, K = 0.7184^(1/17) = 0.9807 again.

Answer 2

The number of cases is 199000 / 277000 = 0.7184 times that of 17 years ago.

Thus your formula is 277000*(0.7184)^(t/17)
= 277000*(0.7184^(1/17))^t
= 277000 *(0.9807)^t.