Question

Hi everyone, so I was doing some math homework and I got stuck. Please help me with these two questions (numbers 7 and 8). Thank you!

I'll also try my hardest to help you in return!

Answer 1

7). x = 6,
`(14)/(3)`

8). x = 2

Explanation:

7). Area of a circle = πr²

Here 'r' = radius of the circle

Area of the shaded region of the circle = Area of the large circle - Area of the small circle

= π(x)² -
`\pi ((x)/(2))^2`

=
`\pi x^(2) -\pi ((x^2)/(4))`

=
`\pi x^(2)(1-(1)/(4))`

=
`(3\pi x^(2) )/(4)` in²

If area of the shaded region is (8πx - 21π) in²

`(3\pi x^(2) )/(4)=8\pi x - 21\pi`

`(3x^2)/(4)=8x-21`

3x² = 32x - 84

3x²- 32x + 84 = 0

3x² - 14x - 18x + 84 = 0

x(3x - 14) - 6(3x - 14) = 0

(x - 6)(3x - 14) = 0

x = 6,
`(14)/(3)`

8). Area of the shaded region = Area of the largest circle - Area of the smaller semicircle - Area of the smallest semicircle

Area of the largest circle = π(2x + 2x)²

= 16πx²

Area of the smaller semicircle =
`(1)/(2)\pi (3x)^(2)`

= 4.5(πx²)

Area of the smallest semicircle =
`(1)/(2)\pi (2x)^(2)`

= 2πx²

Now area of he shaded region = 16πx² - [4.5(πx²) + 2πx²]

= 16πx² - 6.5πx²

= 9.5πx² mm²

If the area of the shaded region = 12πx + 14π

12πx + 14π = 9.5πx²

12x + 14 = 9.5x²

9.5x² - 12x - 14 = 0

19x² - 24x - 28 = 0

19x² - 38x + 14x - 28 = 0

19x(x - 2) + 14(x - 2) = 0

(19x + 14)(x - 2) = 0

x =
`-(14)/(19),2`

But the radius o a circle can't be negative.

Therefore, x = 2 will be the answer.