Question

The number? H(t) of veterans from foreign wars who are homeless or at risk of becoming homeless can be approximated by the exponential? function,

H(t)=65(1.65)^t where t is the number of years since 2000.

?a) In what year were there 14,000 veterans who were homeless or at risk of becoming? homeless?

?b) What is the doubling time of homelessness among? veterans?

Answer 1

Given exponential function for veterans from foreign wars who are homeless or at risk of becoming homeless is :

H(t)=65(1.65)^t where H(t) is the number of such veterans and t is the number of years since 2000.

a) Given, H(t) = 14000

Putting this in the given exponential function we get,

14000 = 65(1.65)^t

i.e., (1.65)^t = 14000/65

i.e., log(1.65)^t = log(14000/65)

i.e., t*log(1.65) = log(14000/65)

i.e., t = [log(14000/65)]/[log(1.65)]

i.e., t = 10.72821577

i.e., t 11

Therefore, in the year 2011 there will be 14,000 veterans who are homeless or at risk of becoming homeless.

b) Given, H(t) = 28000

Putting this in the given exponential function we get,

28000 = 65(1.65)^t

i.e., (1.65)^t = 28000/65

i.e., log(1.65)^t = log(28000/65)

i.e., t*log(1.65) = log(28000/65)

i.e., t = [log(28000/65)]/[log(1.65)]

i.e., t = 12.1123639

Therefore within year 2013, the number of homeless or at risk of becoming homeless veterans will be doubled.

In other words, the doubling time of homelessness among veterans is nearly 13 years.