The area of a tiled kitchen floor is represented by the expression $4 x^2+24 x+27$, where $x$ represents the length of a single tile. The length, $l$, of the floor is 9 feet more than twice the length of a tile, $x$. Which expression represents the width of the kitchen in terms of $x$?
The area of the rectangular kitchen floor is given by the expression \(A(x) = 4x^2 + 24x + 27\), where \(x\) is the length of a single tile. We are also given the length of the kitchen floor, \(l(x) = 2x + 9\). Since the area of a rectangle is the product of its length and its width (\(A = l \times w\)), we can express the width \(w(x)\) as the quotient of the area and the length:
\[w(x) = \frac{A(x)}{l(x)} = \frac{4x^2 + 24x + 27}{2x + 9}\]
To find the width \(w(x)\), we need to divide the polynomial \(4x^2 + 24x + 27\) by \(2x + 9\). We can perform this division by factoring the quadratic expression. We look for two numbers that multiply to \(4 \times 27 = 108\) and add to \(24\). These numbers are \(6\) and \(18\). Thus, we can rewrite the expression as:
\[4x^2 + 24x + 27 = 4x^2 + 6x + 18x + 27\]
Grouping the terms, we get:
\[(4x^2 + 6x) + (18x + 27) = 2x(2x + 3) + 9(2x + 3)\]
Factoring out the common term \((2x + 3)\), we obtain:
\[(2x + 9)(2x + 3)\]
Now, we can substitute this back into our expression for the width:
\[w(x) = \frac{(2x + 9)(2x + 3)}{2x + 9}\]
Dividing both numerator and denominator by \((2x + 9)\), we find the width to be:
\[w(x) = 2x + 3\]
Based on our calculations, the width of the kitchen is \(2x + 3\). Thus, the final expression for the width of the kitchen floor in terms of the tile length \(x\) is \(\boxed{2x + 3}\).
Understanding how to relate area to dimensions is vital in fields like interior design and construction. For example, if you are planning to tile a kitchen floor and know the total area available and one dimension (like the length), you can easily determine the required width. This algebraic skill helps in ordering the correct amount of materials and ensuring that your space is accurately measured before starting any renovation projects.