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 364π cm². what is the value of x? enter your answer in the box. x = cm

Explanation:

Step1: Find area formulas

The area of a circle is $A = \pi r^{2}$. The area of the outer - circle $A_{1}=\pi(2x)^{2}=4\pi x^{2}$, and the area of the inner - circle $A_{2}=\pi\times6^{2}=36\pi$.

Step2: Set up equation for shaded area

The area of the shaded region is the difference between the area of the outer - circle and the area of the inner - circle. So $A_{1}-A_{2}=364\pi$. Substitute the area formulas: $4\pi x^{2}-36\pi = 364\pi$.

Step3: Simplify the equation

Divide both sides of the equation $4\pi x^{2}-36\pi = 364\pi$ by $\pi$ to get $4x^{2}-36 = 364$. Then add 36 to both sides: $4x^{2}=364 + 36=400$.

Step4: Solve for $x$

Divide both sides of $4x^{2}=400$ by 4: $x^{2}=100$. Take the square root of both sides. Since $x$ represents a length, we take the positive square root, so $x = 10$.

Answer:

$10$