Question

fernando, an interior decorator, is designing an entryway for a client’s new home. the entryway will include a set of double doors and hardwood moulding. the moulding will be an equal width around the sides and top of the double doors, which will be 60 inches wide and 80 inches tall.
image of double doors with moulding, labeled x in. on sides and top, 60 in. width of doors, 80 in. height of doors
the total area of the double doors and moulding in square inches can be modeled by the expression ((2x + 60)(x + 80)), where (x) is the width of the moulding in inches. this expression can also be written in the form (2x^2 + 220x + 4,800).
what does the quantity (2x^2 + 220x) represent in the expression?

• the area of the double doors in square inches
• the area of the moulding in square inches
• the height of the double doors and moulding together in inches
• the perimeter around the outside of the double doors and moulding in inches

Explanation:

Step1: Analizar la fórmula del área total

El área total del conjunto (puertas + moldura) se modela como \((2x + 60)(x + 80)\), que al expandir da \(2x^2 + 220x + 4800\). El término \(4800\) se obtiene al calcular \(60\times80\), que es el área de las puertas dobles (ya que las puertas son de \(60\) pulgadas de ancho y \(80\) de alto).

Step2: Identificar el significado de \(2x^2 + 220x\)

Si el área total es (área de puertas) + (área de moldura), y el área de puertas es \(60\times80 = 4800\), entonces \(2x^2 + 220x\) debe representar el área de la moldura. Esto se deduce porque al restar el área de las puertas (\(4800\)) del área total (\(2x^2 + 220x + 4800\)), se obtiene el área de la moldura, que es \(2x^2 + 220x\).

Answer:

the area of the moulding in square inches