To determine the reasonable domain and range for the function [tex]\( W(b) = 12b \)[/tex], we need to consider the given conditions: Owen can build up to 10 birdhouses, and each birdhouse requires 12 square feet of wood.
### Domain
The domain represents the possible number of birdhouses, [tex]\( b \)[/tex], Owen can build. Given that Owen can build up to 10 birdhouses, the minimum number of birdhouses he can build is 0 (he builds none), and the maximum is 10. Therefore, the domain [tex]\( D \)[/tex] can be represented as:
[tex]\[ 0 \leq b \leq 10 \][/tex]
### Range
The range represents the total amount of wood [tex]\( W(b) \)[/tex] needed to build [tex]\( b \)[/tex] birdhouses. Since each birdhouse requires 12 square feet of wood, the amount of wood needed if Owen builds 0 birdhouses is:
[tex]\[ W(0) = 12 \times 0 = 0 \][/tex]
And if Owen builds the maximum number of birdhouses (10):
[tex]\[ W(10) = 12 \times 10 = 120 \][/tex]
Therefore, the range [tex]\( R \)[/tex] can be represented as:
[tex]\[ 0 \leq W(b) \leq 120 \][/tex]
Combining the information about the domain and range, the correct choice is:
[tex]\[ C: \, D : 0 \leq b \leq 10, \, R: 0 \leq W(b) \leq 120 \][/tex]
