let

x = the number of breast cancer deaths in 1990

39,620 = the number of breast cancer deaths in 2013

the rate has dropped by 34% since 1990 that is

x - 0.34x = 39,620

0,66 x = 39,620

x = 39,620/0.66

a. Given the 34% decrease in breast cancer deaths from 1990 to 2013, find the yearly rate of decrease (as a percent) in breast cancer deaths.

let i = the yearly rate of decrease or the decay factor

in the year of 1990

x = 39,620/0.66 deaths

in the year of 1991

x - i*x = (1-i)x

in the year of 1992

(1-i)x - i(1-i)x = (1-i)^2*x

in the year of 2013

(1-i)^23*x = 39620

(1-i)^23 = 39620/x

(1-i)^23 = 39620/(39,620/0.66)

(1-i)^23 = 0.66

1 - i = 0.9685

i = 1 - 0.9685

i = 0.0315

i = 3.15%

the yearly rate of decrease (as a percent) in breast cancer deaths is 3.15%

Find an exponential model to predict the number of breast cancer deaths D, in terms of the number of years t since 2013.

D(t) = C*a^t, where t is the number of years since 2013

D(0) = 39620, the number of deaths in the year of 2013 thus

C*a^0 = 39620

C = 39620

D(t) = 39620*a^t, a is the decay factor

since i is the yearly rate of decrease(the decay rate), the decay factor a = 1 - i = 1 - 0.0315 = 0.9685 thus

D(t) = 39620*(0.9685)^t is the exponential model to predict the number of breast cancer deaths.