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The width of a rectangle is modeled by the function [tex]\( w(x) = 3x - 6 \)[/tex], and the length of the rectangle is modeled by the function [tex]\( l(x) = 5x + 7 \)[/tex].

How can the functions be combined to find a function, [tex]\( a(x) \)[/tex], that models the area of the rectangle?

A. [tex]\( a(x) = 8x + 1 \)[/tex]
B. [tex]\( a(x) = 16x + 2 \)[/tex]
C. [tex]\( a(x) = 15x^2 - 9x - 42 \)[/tex]
D. [tex]\( a(x) = 30x^2 - 18x - 84 \)[/tex]

Sagot :

GinnyAnswer
To find the function [tex]\(a(x)\)[/tex] that models the area of the rectangle, we need to combine the given functions for width [tex]\(w(x)\)[/tex] and length [tex]\(l(x)\)[/tex].

1. The width of the rectangle is given by:
[tex]\[ w(x) = 3x - 6 \][/tex]

2. The length of the rectangle is given by:
[tex]\[ l(x) = 5x + 7 \][/tex]

3. The area of the rectangle [tex]\(a(x)\)[/tex] is found by multiplying the width by the length:
[tex]\[ a(x) = w(x) \times l(x) \][/tex]

4. Substitute the expressions for [tex]\(w(x)\)[/tex] and [tex]\(l(x)\)[/tex]:
[tex]\[ a(x) = (3x - 6) \times (5x + 7) \][/tex]

5. Expand the multiplication using the distributive property (also known as FOIL for binomials):
[tex]\[ (3x - 6)(5x + 7) = 3x \cdot 5x + 3x \cdot 7 - 6 \cdot 5x - 6 \cdot 7 \][/tex]

6. Perform the individual multiplications:
[tex]\[ = 15x^2 + 21x - 30x - 42 \][/tex]

7. Combine like terms:
[tex]\[ 15x^2 + 21x - 30x - 42 = 15x^2 - 9x - 42 \][/tex]

Therefore, the function that models the area of the rectangle is:
[tex]\[ a(x) = 15x^2 - 9x - 42 \][/tex]

So, the correct answer is:
[tex]\[ \boxed{15x^2 - 9x - 42} \][/tex]
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