The cost of fighting crime in the U.S. increased steadily in the period 1982 - 1999. Total spending on police, courts, and prisons can be approximated, respectively, by
$P ( t ) = 1.745 t + 29.84$

billion dollars $( 2 \leq t \leq 19 )$ $C ( t ) = 1.097 t + 10.65$ billion dollars $( 2 \leq t \leq 19 )$ $J ( t ) = 1.919 t + 12.36$ billion dollars $( 2 \leq t \leq 19 )$
Where t is time in years since 1980. Compute $\lim _ { t \rightarrow \infty } \frac { P ( t ) } { P ( t ) + C ( t ) + J ( t ) }$ to two decimal places.

A)0.27
B) 0.37
C) 0.32
D) 0.42
E) 0.52

Question

The cost of fighting crime in the U.S. increased steadily in the period 1982 - 1999. Total spending on police, courts, and prisons can be approximated, respectively, by  
 $P ( t ) = 1.745 t + 29.84$

billion dollars $( 2 \leq t \leq 19 )$ $C ( t ) = 1.097 t + 10.65$ billion dollars $( 2 \leq t \leq 19 )$ $J ( t ) = 1.919 t + 12.36$ billion dollars $( 2 \leq t \leq 19 )$  
Where t is time in years since 1980. Compute $\lim _ { t \rightarrow \infty } \frac { P ( t ) } { P ( t ) + C ( t ) + J ( t ) }$ to two decimal places.
 
A)0.27
B) 0.37
C) 0.32
D) 0.42
E) 0.52

Quizplus · Accepted Answer

To find the limit as $t \rightarrow \infty$ of $\frac{P(t)}{P(t) + C(t) + J(t)}$, we first sum the functions in the denominator and then divide the numerator by the sum. The functions are linear, so their coefficients will determine the behavior as $t$ approaches infinity. The limit simplifies to the ratio of the coefficients of $t$ in $P(t)$ over the sum of the coefficients of $t$ in $P(t)$, $C(t)$, and $J(t)$. Thus, $\lim _ { t \rightarrow \infty } \frac { P ( t ) } { P ( t ) + C ( t ) + J ( t ) } = \frac{1.745}{1.745 + 1.097 + 1.919} = \frac{1.745}{4.761}$. Calculating this gives approximately 0.37.

The Cost of Fighting Crime in the U P(t)=1.745t+29.84P ( t ) = 1.745 t + 29.84P(t)=1.745t+29.84

The Cost of Fighting Crime in the U $P ( t ) = 1.745 t + 29.84$