Certainly! Let's tackle both parts of the question step by step.
### Part (a) Simplifying the Expression [tex]\( C(x) \)[/tex]
The given function is:
[tex]\[ C(x) = \frac{9040}{33(51 - x)} - \frac{8}{33} \][/tex]
First, perform the indicated subtraction. To do this, we need a common denominator.
The common denominator of the two fractions:
[tex]\[ \text{Common Denominator} = 33(51 - x) \][/tex]
Rewrite both fractions with this common denominator:
[tex]\[ \frac{9040}{33(51 - x)} = \frac{9040}{33(51 - x)} \][/tex]
[tex]\[ \frac{8}{33} = \frac{8(51 - x)}{33(51 - x)} = \frac{408 - 8x}{33(51 - x)} \][/tex]
Now, subtract these fractions:
[tex]\[ C(x) = \frac{9040}{33(51 - x)} - \frac{408 - 8x}{33(51 - x)} \][/tex]
Since both fractions have the same denominator, combine the numerators over the common denominator:
[tex]\[ C(x) = \frac{9040 - (408 - 8x)}{33(51 - x)} = \frac{9040 - 408 + 8x}{33(51 - x)} \][/tex]
Simplify the numerator:
[tex]\[ 9040 - 408 = 8632 \][/tex]
[tex]\[ \text{So, we have:} \][/tex]
[tex]\[ C(x) = \frac{8632 + 8x}{33(51 - x)} \][/tex]
Factor out an 8 from the numerator:
[tex]\[ C(x) = \frac{8(1079 + x)}{33(51 - x)} \][/tex]
Simplify further:
[tex]\[ C(x) = \frac{-0.242424242424242x - 261.575757575758}{x - 51} \][/tex]
### Part (b) Calculate the cost for 38 points
With the simplified expression:
[tex]\[ C(x) = \frac{-0.242424242424242x - 261.575757575758}{x - 51} \][/tex]
Substitute [tex]\( x = 38 \)[/tex] into this expression to find the cost at 38 points:
[tex]\[ C(38) = \frac{-0.242424242424242(38) - 261.575757575758}{38 - 51} \][/tex]
First, compute the numerator:
[tex]\[ -0.242424242424242 \times 38 = -9.212121212121212 \][/tex]
[tex]\[ -9.212121212121212 - 261.575757575758 = -270.7878787878792 \][/tex]
Then compute the denominator:
[tex]\[ 38 - 51 = -13 \][/tex]
Now, divide the numerator by the denominator:
[tex]\[ C(38) = \frac{-270.7878787878792}{-13} \approx 20.83 \][/tex]
So, the cost to win 38 points, rounded to two decimal places, is approximately:
[tex]\[ \boxed{20.83 \text{ thousand dollars}} \][/tex]
Thus, it would cost around 20.83 thousand dollars to win 38 points.
