To determine the grooming cost for a 60-pound dog using the function [tex]\( f(x) \)[/tex], we need to evaluate which case from the piecewise function applies:
[tex]\[ f(x) = \left\{
\begin{array}{cc}
45, & 0 < x \leq 12 \\
55, & 12 < x \leq 45 \\
55 + 3(x - 45), & x > 45
\end{array}
\right. \][/tex]
Given the weight of the dog, [tex]\(x = 60\)[/tex].
1. Check the conditions for [tex]\( f(x) \)[/tex]:
- [tex]\( 0 < x \leq 12 \)[/tex]: This does not apply, since [tex]\( x = 60 \)[/tex].
- [tex]\( 12 < x \leq 45 \)[/tex]: This does not apply, since [tex]\( x = 60 \)[/tex].
2. Apply the condition for [tex]\( x > 45 \)[/tex]:
- Since [tex]\( x = 60 \)[/tex] is greater than 45, we use the formula:
[tex]\[ f(x) = 55 + 3(x - 45) \][/tex]
3. Substitute [tex]\( x = 60 \)[/tex] into the expression:
[tex]\[ f(60) = 55 + 3(60 - 45) \][/tex]
[tex]\[ f(60) = 55 + 3(15) \][/tex]
[tex]\[ f(60) = 55 + 45 \][/tex]
[tex]\[ f(60) = 100 \][/tex]
Thus, the grooming cost for a 60-pound dog is:
[tex]\[ \boxed{100} \][/tex]
