9. Find the diameter of a circle with an area of 415.48 square inches.

To find the diameter of a circle given its area, we can use the formula for the area of a circle: \[ A = \pi r^2 \] where: - \( A \) is the area of the circle, - \( r \) is the radius of the circle. Given that the area of the circle is 415.48 square inches, we can set up the equation: \[ 415.48 = \pi r^2 \] Now, we can solve for the radius \( r \) and then find the diameter of the circle using the formula: \[ \text{Diameter} = 2r \] Let's solve for the radius first. Solve the quadratic equation by following steps: - step0: Solve using square roots: \(415.48=\pi r^{2}\) - step1: Convert the decimal into a fraction: \(\frac{10387}{25}=\pi r^{2}\) - step2: Swap the sides: \(\pi r^{2}=\frac{10387}{25}\) - step3: Divide both sides: \(\frac{\pi r^{2}}{\pi }=\frac{\frac{10387}{25}}{\pi }\) - step4: Divide the numbers: \(r^{2}=\frac{10387}{25\pi }\) - step5: Simplify the expression: \(r=\pm \sqrt{\frac{10387}{25\pi }}\) - step6: Simplify the expression: \(r=\pm \frac{\sqrt{10387\pi }}{5\pi }\) - step7: Separate into possible cases: \(\begin{align}&r=\frac{\sqrt{10387\pi }}{5\pi }\\&r=-\frac{\sqrt{10387\pi }}{5\pi }\end{align}\) - step8: Rewrite: \(r_{1}=-\frac{\sqrt{10387\pi }}{5\pi },r_{2}=\frac{\sqrt{10387\pi }}{5\pi }\) The solutions for the radius \( r \) are: \[ r_{1} = -\frac{\sqrt{10387\pi}}{5\pi} \] \[ r_{2} = \frac{\sqrt{10387\pi}}{5\pi} \] Since the radius cannot be negative, we take the positive solution: \[ r = \frac{\sqrt{10387\pi}}{5\pi} \] Now, we can find the diameter of the circle using the formula: \[ \text{Diameter} = 2r \] Substitute the value of \( r \) into the formula to find the diameter. Calculate the value by following steps: - step0: Calculate: \(2\times \frac{\sqrt{10387\pi }}{5\pi }\) - step1: Multiply the fractions: \(\frac{2\sqrt{10387\pi }}{5\pi }\) The diameter of the circle is approximately 23 inches.