Question

a pool measuring 14 meters by 20 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 720 square meters, what is the width of the path?
the width of the path is \\(\square\\) meters.

Explanation:

Step1: Set up area equation

The combined area is the area of the larger rectangle, so:
$$(14+2x)(20+2x)=720$$

Step2: Expand the left side

Multiply the two binomials:
$$280 + 28x + 40x + 4x^2 = 720$$
$$4x^2 + 68x + 280 = 720$$

Step3: Simplify to standard quadratic

Subtract 720 from both sides, then divide by 4:
$$4x^2 + 68x - 440 = 0$$
$$x^2 + 17x - 110 = 0$$

Step4: Factor the quadratic

Find two numbers that multiply to -110 and add to 17:
$$(x+22)(x-5)=0$$

Step5: Solve for x

Set each factor equal to 0, discard negative width:
$$x+22=0 \implies x=-22 \ (\text{invalid})$$
$$x-5=0 \implies x=5$$

Answer:

5