To calculate \((f \cdot g)(x)\), we first find the product of the two functions \(f(x)\) and \(g(x)\). The functions are defined as:
- \(f(x) = x\) (the length of the sandbox in yards)
- \(g(x) = x + 6\) (the width of the sandbox in yards)
Now, we multiply the two functions:
\[ (f \cdot g)(x) = f(x) \cdot g(x) = x \cdot (x + 6) \]
Now, let's simplify this expression:
\[ (f \cdot g)(x) = x(x + 6) = x^2 + 6x \]
So, \((f \cdot g)(x) = x^2 + 6x\).
Situation Described by \((f \cdot g)(x)\)
The expression \((f \cdot g)(x) = x^2 + 6x\) represents the area of the sandbox in square yards, where \(x\) is the length of the sandbox in yards and \(g(x) = x + 6\) is the width.
In this scenario:
- \(x\) is the length of the sandbox,
- \(x + 6\) is the width of the sandbox,
- The area of the sandbox is calculated by multiplying the length and width together, resulting in the expression \(x^2 + 6x\).
Thus, \((f \cdot g)(x)\) describes the area of a rectangular sandbox where the length is \(x\) yards and the width is \(x + 6\) yards.