To determine the difference in the number of customers visiting Mia's coffee shop and ice cream parlor, we need to calculate the difference function [tex]\(D(x)\)[/tex], which is given by:
[tex]\[ D(x) = W(x) - R(x) \][/tex]
where:
[tex]\[ W(x) = 0.002x^3 - 0.01x^2 \][/tex]
[tex]\[ R(x) = x^2 - 4x + 13 \][/tex]
Now we need to subtract the function [tex]\(R(x)\)[/tex] from the function [tex]\(W(x)\)[/tex]:
[tex]\[ D(x) = W(x) - R(x) \][/tex]
[tex]\[ D(x) = (0.002x^3 - 0.01x^2) - (x^2 - 4x + 13) \][/tex]
Next, we distribute the subtraction:
[tex]\[ D(x) = 0.002x^3 - 0.01x^2 - x^2 + 4x - 13 \][/tex]
Now, we combine the like terms. Begin with the terms involving [tex]\(x^2\)[/tex]:
[tex]\[ 0.002x^3 - (0.01x^2 + x^2) + 4x - 13 \][/tex]
Combine [tex]\( -0.01x^2 \)[/tex] and [tex]\( -x^2 \)[/tex]:
[tex]\[ -0.01x^2 - x^2 = -1.01x^2 \][/tex]
Thus, the equation becomes:
[tex]\[ D(x) = 0.002x^3 - 1.01x^2 + 4x - 13 \][/tex]
Therefore, the simplified function representing the difference in the number of customers visiting the two shops is:
[tex]\[ D(x) = 0.002x^3 - 1.01x^2 + 4x - 13 \][/tex]
The correct choice is:
C. [tex]\( D(x) = 0.002 x^3 - 1.01 x^2 + 4 x - 13 \)[/tex]
