a rectangular deck is 12 ft by 14 ft. when the length and width are increased by the same amount, the area…

a rectangular deck is 12 ft by 14 ft. when the length and width are increased by the same amount, the area becomes 288 sq. ft. by how much were the dimensions increased?\n2 ft\n4 ft\n8 ft\n10 ft

a rectangular deck is 12 ft by 14 ft. when the length and width are increased by the same amount, the area becomes 288 sq. ft. by how much were the dimensions increased?\n2 ft\n4 ft\n8 ft\n10 ft

Answer

Explanation:

Step1: Definir la variable

Sea $x$ la cantidad por la que se aumentan la longitud y el ancho.

Step2: Escribir la ecuación del área

La nueva longitud es $(14 + x)$ pies y el nuevo ancho es $(12 + x)$ pies. El área $A=(14 + x)(12 + x)$. Sabemos que $A = 288$ pies cuadrados. Entonces, $(14 + x)(12 + x)=288$.

Step3: Expandir la ecuación

Expandiendo $(14 + x)(12 + x)$ obtenemos $168+14x + 12x+x^{2}=288$, que se simplifica a $x^{2}+26x+168 - 288=0$, es decir, $x^{2}+26x - 120 = 0$.

Step4: Resolver la ecuación cuadrática

Factorizamos $x^{2}+26x - 120=(x + 30)(x - 4)=0$.

Step5: Encontrar los valores de $x$

Si $x+30 = 0$, entonces $x=-30$; si $x - 4=0$, entonces $x = 4$. Descartamos $x=-30$ ya que no puede ser una cantidad negativa de aumento.

Answer:

4 ft