Answer:
7). x = 6, [tex]\frac{14}{3}[/tex]
8). x = 2
Step-by-step 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 - [tex]\pi (\frac{x}{2})^2[/tex]
= [tex]\pi x^{2} -\pi (\frac{x^2}{4})[/tex]
= [tex]\pi x^{2}(1-\frac{1}{4})[/tex]
= [tex]\frac{3\pi x^{2} }{4}[/tex] in²
If area of the shaded region is (8πx - 21π) in²
[tex]\frac{3\pi x^{2} }{4}=8\pi x - 21\pi[/tex]
[tex]\frac{3x^2}{4}=8x-21[/tex]
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, [tex]\frac{14}{3}[/tex]
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 = [tex]\frac{1}{2}\pi (3x)^{2}[/tex]
= 4.5(πx²)
Area of the smallest semicircle = [tex]\frac{1}{2}\pi (2x)^{2}[/tex]
= 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 = [tex]-\frac{14}{19},2[/tex]
But the radius o a circle can't be negative.
Therefore, x = 2 will be the answer.
