Question

in the diagram, the radius of the outer circle is 2x cm and the radius of the inside circle is 6 cm. the area of the shaded region is $220pi\\ \mathrm{cm}^2$. what is the value of x? enter your answer in the box. x = \square cm

Explanation:

Step1: Set up area difference equation

The area of the shaded region is the area of the outer circle minus the area of the inner circle. The area of a circle is $\pi r^2$, so:
$$\pi(2x)^2 - \pi(6)^2 = 220\pi$$

Step2: Simplify the equation

Divide all terms by $\pi$ to eliminate it, then expand and simplify:
$$(2x)^2 - 6^2 = 220$$
$$4x^2 - 36 = 220$$

Step3: Isolate the $x^2$ term

Add 36 to both sides of the equation:
$$4x^2 = 220 + 36$$
$$4x^2 = 256$$

Step4: Solve for $x^2$

Divide both sides by 4:
$$x^2 = \frac{256}{4}$$
$$x^2 = 64$$

Step5: Solve for $x$

Take the positive square root (since radius is positive):
$$x = \sqrt{64}$$
$$x = 8$$

Answer:

$x = 8$ cm