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A company sells bikes that are light and fast. The revenue earned over a period of [tex][tex]$x$[/tex][/tex] years can be represented by the function [tex][tex]$S(x)=100 x^3+6 x^2+97 x+215$[/tex][/tex]. The cost of manufacturing the bikes can be represented by the function [tex][tex]$C(x)=88 x+215$[/tex][/tex].

Which function [tex][tex]$P(x)$[/tex][/tex] describes the profit earned from the sales of the bikes over a period of [tex][tex]$x$[/tex][/tex] years?

A. [tex][tex]$P(x)=100 x^3+6 x^2+9 x$[/tex][/tex]

B. [tex][tex]$P(x)=100 x^2+6 x+9$[/tex][/tex]

C. [tex][tex]$P(x)=100 x^3+6 x^2+185 x+430$[/tex][/tex]

D. [tex][tex]$P(x)=100 x^3-82 x^2+97 x$[/tex][/tex]



Answer :

GinnyAnswer
To find the profit function, [tex]\( P(x) \)[/tex], we need to calculate the difference between the revenue function, [tex]\( S(x) \)[/tex], and the cost function, [tex]\( C(x) \)[/tex].

The given functions are:
[tex]\[ S(x) = 100x^3 + 6x^2 + 97x + 215 \][/tex]
[tex]\[ C(x) = 88x + 215 \][/tex]

The profit function [tex]\( P(x) \)[/tex] is defined as:
[tex]\[ P(x) = S(x) - C(x) \][/tex]

Now, let’s perform the subtraction step-by-step:

1. Write down the revenue function [tex]\( S(x) \)[/tex] and the cost function [tex]\( C(x) \)[/tex]:
[tex]\[ S(x) = 100x^3 + 6x^2 + 97x + 215 \][/tex]
[tex]\[ C(x) = 88x + 215 \][/tex]

2. Subtract [tex]\( C(x) \)[/tex] from [tex]\( S(x) \)[/tex]:
[tex]\[ P(x) = (100x^3 + 6x^2 + 97x + 215) - (88x + 215) \][/tex]

3. Distribute the negative sign and combine like terms:
[tex]\[ P(x) = 100x^3 + 6x^2 + 97x + 215 - 88x - 215 \][/tex]

4. Combine the constants and like terms:
[tex]\[ P(x) = 100x^3 + 6x^2 + (97x - 88x) + (215 - 215) \][/tex]
[tex]\[ P(x) = 100x^3 + 6x^2 + 9x \][/tex]

Therefore, the correct profit function [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = 100x^3 + 6x^2 + 9x \][/tex]

The correct answer is:
A. [tex]\( P(x) = 100x^3 + 6x^2 + 9x \)[/tex]

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