Question

a pool measuring 12 meters by 30 meters is surrounded by a path of uniform width, as shown in the figure. if the area of the pool and the path combined is 1144 square meters, what is the width of the path? the width of the path is meters.

Explanation:

Step1: Find area formula of pool + path

The outer - rectangle (pool + path) has length $l = 30 + 2x$ and width $w=12 + 2x$. The area formula for a rectangle is $A=l\times w$. So the area of the pool and path combined is $A=(30 + 2x)(12 + 2x)$.

Step2: Expand the expression

Using the FOIL method:

$$
LATEXBLOCK0
$$

Step3: Set up the equation

We know that the area of the pool and path combined is 1144 square meters. So we set up the equation $4x^{2}+84x + 360=1144$.

Step4: Rearrange the equation to standard quadratic form

Subtract 1144 from both sides: $4x^{2}+84x+360 - 1144 = 0$, which simplifies to $4x^{2}+84x - 784 = 0$. Divide through by 4 to get $x^{2}+21x - 196 = 0$.

Step5: Solve the quadratic equation

We can factor the quadratic equation $x^{2}+21x - 196=(x + 28)(x - 7)=0$.
Setting each factor equal to zero gives $x+28 = 0$ or $x - 7=0$. So $x=-28$ or $x = 7$. Since the width cannot be negative, we discard $x=-28$.

Answer:

7