Certainly! Let's go through the problems step by step.
### Part (a): Simplify the expression for [tex]\( C(x) \)[/tex]
The given function for the cost, in thousands of dollars, of restoring a car to win [tex]\( x \)[/tex] points is:
[tex]\[
C(x) = \frac{9040}{22(51 - x)} - \frac{9}{22}
\][/tex]
First, we simplify each term separately.
#### Step 1: Simplify the constant term [tex]\(\frac{9}{22}\)[/tex]
[tex]\[
\frac{9}{22} \approx 0.4090909090909091
\][/tex]
#### Step 2: Simplify the coefficient in the numerator [tex]\(\frac{9040}{22}\)[/tex]
[tex]\[
\frac{9040}{22} \approx 410.90909090909093
\][/tex]
Putting it all together, we get the simplified expression for [tex]\( C(x) \)[/tex]:
[tex]\[
C(x) = \frac{410.90909090909093}{51 - x} - 0.4090909090909091
\][/tex]
### Part (b): Determine the cost to win 42 points
To find the cost for [tex]\( x = 42 \)[/tex] points, substitute [tex]\( x = 42 \)[/tex] into the simplified expression:
[tex]\[
C(42) = \frac{410.90909090909093}{51 - 42} - 0.4090909090909091
\][/tex]
#### Step 1: Calculate the denominator
[tex]\[
51 - 42 = 9
\][/tex]
#### Step 2: Substitute the denominator into the simplified expression
[tex]\[
C(42) = \frac{410.90909090909093}{9} - 0.4090909090909091
\][/tex]
#### Step 3: Perform the division
[tex]\[
\frac{410.90909090909093}{9} \approx 45.65656767878788
\][/tex]
#### Step 4: Subtract the constant term
[tex]\[
45.65656767878788 - 0.4090909090909091 \approx 45.24747676969697
\][/tex]
Rounding this to two decimal places, we get:
[tex]\[
C(42) \approx 45.25
\][/tex]
So, it would cost approximately \$45,250 to win 42 points in the competition.
