Let's go through the steps to solve Rachael's problem.
### Part (a): Find the polynomial that represents the profit for Rachael's business
To find the profit, we need to subtract the cost from the revenue. We are given:
- Cost: [tex]\( C(x) = 2000 + 2x \)[/tex]
- Revenue: [tex]\( R(x) = 12x - 0.0005x^2 \)[/tex]
The profit, [tex]\( P(x) \)[/tex], is given by:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given polynomials:
[tex]\[ P(x) = (12x - 0.0005x^2) - (2000 + 2x) \][/tex]
Now, simplify the expression:
[tex]\[ P(x) = 12x - 0.0005x^2 - 2000 - 2x \][/tex]
Combine like terms:
[tex]\[ P(x) = (12x - 2x) - 0.0005x^2 - 2000 \][/tex]
[tex]\[ P(x) = 10x - 0.0005x^2 - 2000 \][/tex]
So, the polynomial that represents the profit is:
[tex]\[ P(x) = 10x - 0.0005x^2 - 2000 \][/tex]
### Part (b): Calculate Rachael's profit if she sells 5,000 batches of cookies a month
To find the profit for 5,000 batches, we need to evaluate the profit polynomial at [tex]\( x = 5000 \)[/tex].
So, substitute [tex]\( x = 5000 \)[/tex] into the polynomial:
[tex]\[ P(5000) = 10(5000) - 0.0005(5000)^2 - 2000 \][/tex]
Now, calculate each term:
[tex]\[ 10(5000) = 50000 \][/tex]
[tex]\[ 0.0005(5000)^2 = 0.0005 \times 25000000 = 12500 \][/tex]
Substitute these values back into the equation:
[tex]\[ P(5000) = 50000 - 12500 - 2000 \][/tex]
[tex]\[ P(5000) = 50000 - 14500 \][/tex]
[tex]\[ P(5000) = 35500 \][/tex]
Thus, Rachael's profit from selling 5,000 batches of cookies a month is:
[tex]\[ \boxed{35500 \text{ dollars}} \][/tex]
