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9514 1404 393

  x = 1/2

Step-by-step explanation:

There are some different ways to find the area of a donut shape. One is to multiply its centerline length by its width. We'll use that here to find the shaded area of the sector.

The centerline radius of shaded area A is ...

  (6x -2) cm + (1/2)(4x cm) = 8x -2 cm

The length of the centerline is ...

  s = rθ = (8x -2)(π/4) . . . . . . for a central angle of 45° = π/4 radians

Then the area A is ...

  area A = (π/4)(8x -2)(4x) = 2πx(4x -1) . . . cm^2

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The radius of circle B is ((4 -4x) cm)/2 = (2 -2x) cm. The area of circle B is ...

  A = πr^2 = π(2 -2x)^2 cm^2

We're told the two areas are equal, so we have ...

  2πx(4x -1) = π(2 -2x)^2 . . . . . . . expressions for the areas are equal

  8x^2 -2x = 4x^2 -8x +4 . . . . . . divide by π, eliminate parentheses

  4x^2 +6x -4 = 0 . . . . . . . . . . . . subtract the right-side expression

  2(2x -1)(x +2) = 0 . . . . . . . . . . . factor

  x = 1/2 or -2 . . . . . . . only the positive solution is useful here

The value of x is 1/2.

hi

step-by-step explanation: