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Answer:
x = 5.00 cm
Step-by-step explanation:
• area of the rectangle = [tex]x[/tex] x [tex](x + 1)[/tex]
= [tex]x^2 + x[/tex]
• area of the circle = π r²
= [tex]\frac{22}{7}[/tex] x [tex](x + 2)^2[/tex]
= [tex]\frac{22}{7}[/tex] x [tex](x^2 + 4x + 4)[/tex]
∴ [tex]x^2 + x[/tex] + [tex]\frac{22}{7}[/tex] x [tex](x^2 + 4x + 4) = 184[/tex]
⇒ [tex]x^2 + x[/tex] + [tex]\frac{22}{7} x^2 \space\ + \space\ \frac{88}{7} x \space\ + \space\ \frac{88}{7} = 184[/tex]
⇒ [tex]\frac{29}{7}x^2 \space\ + \space\ \frac{95}{7}x \space\ - \frac{1200}{7} = 0[/tex]
• Using quadratic formula, where
a = [tex]\frac{29}{7}[/tex]
b = [tex]\frac{95}{7}[/tex]
c = [tex]\frac{-1200}{7}[/tex] ,
[tex]x=\frac{-b \pm \sqrt{b^2-4c}}{2}[/tex]
[tex]x = \frac{-\frac{95}{7}\pm \sqrt{(\frac{95}{7})^2 - 4(\frac{29}{7})(\frac{-1200}{7} ) } }{2(\frac{29}{7})}[/tex]
[tex]x = 5.00 \space\ \space\ \space\ or \space\ \space\ \space\ x = -8.28[/tex]
As length (x) cannot be a negative number,
x = 5.00 cm