Let's break down the problem step-by-step.
### Part (a)
We are given two equations:
[tex]\[ D = 2n + 3 \][/tex]
[tex]\[ C = n + 4 \][/tex]
where [tex]\( D \)[/tex] represents the number of dogs adopted (in hundreds), and [tex]\( C \)[/tex] represents the number of cats adopted (in hundreds). The variable [tex]\( n \)[/tex] is the number of years since 1999.
We are asked to write a function that models the total number [tex]\( T \)[/tex] of dogs and cats adopted in hundreds. The total number of dogs and cats adopted [tex]\( T \)[/tex] can be found by summing [tex]\( D \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ T(n) = D + C \][/tex]
Substitute the given equations for [tex]\( D \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ T(n) = (2n + 3) + (n + 4) \][/tex]
Combine like terms:
[tex]\[ T(n) = 2n + n + 3 + 4 \][/tex]
[tex]\[ T(n) = 3n + 7 \][/tex]
Thus, the function that models the total number of dogs and cats adopted in hundreds is:
[tex]\[ T(n) = 3n + 7 \][/tex]
### Part (b)
We need to determine how many dogs and cats will be adopted in 2013 if the trend continues. First, we calculate the number of years since 1999 to 2013. Let [tex]\( n \)[/tex] represent this number of years:
[tex]\[ n = 2013 - 1999 \][/tex]
[tex]\[ n = 14 \][/tex]
Next, we use the function [tex]\( T(n) = 3n + 7 \)[/tex] to find the total number of dogs and cats adopted in 2013 by substituting [tex]\( n = 14 \)[/tex]:
[tex]\[ T(14) = 3(14) + 7 \][/tex]
[tex]\[ T(14) = 42 + 7 \][/tex]
[tex]\[ T(14) = 49 \][/tex]
Therefore, the total number of dogs and cats adopted in 2013 is 49 hundred, or 4900 when expressed in individual units.
### Conclusion
- The function that models the total number of dogs and cats adopted in hundreds is [tex]\( T(n) = 3n + 7 \)[/tex].
- Based on this trend, 49 hundred dogs and cats (or 4900 dogs and cats) will be adopted in 2013.
