Mia opens a coffee shop in the first week of January. The function [tex]W(x) = 0.002x^3 - 0.01x^2[/tex] models the total number of customers visiting the shop since it opened after [tex]x[/tex] days. She then opens an ice cream parlor in the month of February. The function [tex]R(x) = x^2 - 4x + 13[/tex] models the total number of customers visiting the parlor since it opened after [tex]x[/tex] days.

Which function represents the difference, [tex]D(x)[/tex], in the number of customers visiting the two shops?

A. [tex]D(x) = 0.002x^3 + 1.01x^2 - 4x + 13[/tex]
B. [tex]D(x) = 0.002x^3 + 0.99x^2 + 4x - 13[/tex]
C. [tex]D(x) = 0.002x^3 - 1.01x^2 + 4x - 13[/tex]
D. [tex]D(x) = 0.002x^3 - 0.99x^2 - 4x + 13[/tex]



Answer :

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]