Question

The cost of fighting crime in a country increased significantly during the period 1982–1999. Total spending on police, courts, and prisons in the period 1982–1999 could be approximated, respectively, by

P(t) = 1.743t + 29.84 billion dollars (2 ≤ t ≤ 19)

C(t) = 1.096t + 10.65 billion dollars (2 ≤ t ≤ 19)

J(t) = 1.917t + 12.36 billion dollars (2 ≤ t ≤ 19)

where t is time in years since 1980.


Compute lim t→+[infinity]


P(t)

P(t) + C(t) +J(t)

Answer 1

t→+[infinity] [ P ( t ) ] = $62.957 billion

t→+[infinity] [ f ( t ) ] = $143.214 billion

Explanation:

Given:

- Total spending on police, courts, and prisons in the period 1982–1999 could be approximated, respectively,

P(t) = 1.743*t + 29.84 billion dollars (2 ≤ t ≤ 19)

C(t) = 1.096*t + 10.65 billion dollars (2 ≤ t ≤ 19)

J(t) = 1.917*t + 12.36 billion dollars (2 ≤ t ≤ 19)

Find:

- Compute lim t→+[infinity] for:

P ( t ) and P ( t ) + C ( t ) + J ( t )

Solution:

- The limit as t→+[infinity] for the above three function can be accounted for by considering the domain of each function.

- All functions : P ( t ) , C ( t ) , J ( t ) have the domain 2 ≤ t ≤ 19:

- So in other words, lim t→+[infinity] = lim t→19

- The limits are as follows:

lim t→19 [ P ( 19 ) ] = 1.743*19 + 29.84 = $62.957 billion

- The function f ( t ) is as follows:

f( t ) = P ( t ) + C ( t ) + J ( t ) = 4.756*t + 52.85 billion dollars

lim t→19 [ f ( 19 ) ] = 4.756*19 + 52.85 = $143.214 billion