Let's break down the problem step-by-step based on the given piecewise function for the cost of an ad, [tex]\( a(x) \)[/tex]:
[tex]\[
a(x) =
\begin{cases}
45 & \text{ if } x \leq 3 \\
45 + 9(x - 3) & \text{ if } x > 3
\end{cases}
\][/tex]
### Part (a): Difference Between the Cost of a Two-Line Ad and a Three-Line Ad
1. Cost of a Two-Line Ad:
For [tex]\( x = 2 \)[/tex]:
Since [tex]\( x \leq 3 \)[/tex], we use the first part of the piecewise function:
[tex]\[
a(2) = 45
\][/tex]
2. Cost of a Three-Line Ad:
For [tex]\( x = 3 \)[/tex]:
Since [tex]\( x \leq 3 \)[/tex], we use the first part of the piecewise function:
[tex]\[
a(3) = 45
\][/tex]
3. Difference:
The difference between the cost of a two-line ad and a three-line ad is:
[tex]\[
a(3) - a(2) = 45 - 45 = 0
\][/tex]
So, the difference is [tex]\( \boxed{0} \)[/tex].
### Part (b): Cost of a 10-Line Ad
For [tex]\( x = 10 \)[/tex]:
Since [tex]\( x > 3 \)[/tex], we use the second part of the piecewise function:
[tex]\[
a(10) = 45 + 9(10 - 3)
\][/tex]
[tex]\[
a(10) = 45 + 9 \cdot 7
\][/tex]
[tex]\[
a(10) = 45 + 63 = 108
\][/tex]
So, the cost of a 10-line ad is [tex]\( \boxed{108} \)[/tex].
### Part (c): Cost of an 11-Line Ad
For [tex]\( x = 11 \)[/tex]:
Since [tex]\( x > 3 \)[/tex], we use the second part of the piecewise function:
[tex]\[
a(11) = 45 + 9(11 - 3)
\][/tex]
[tex]\[
a(11) = 45 + 9 \cdot 8
\][/tex]
[tex]\[
a(11) = 45 + 72 = 117
\][/tex]
So, the cost of an 11-line ad is [tex]\( \boxed{117} \)[/tex].
In summary, we have:
a. The difference between the cost of a two-line and a three-line ad is [tex]\( \boxed{0} \)[/tex].
b. The cost of a 10-line ad is [tex]\( \boxed{108} \)[/tex].
c. The cost of an 11-line ad is [tex]\( \boxed{117} \)[/tex].
