Plastic Factory A certain machine in a factory fills a spherical mold with plastic. If the diameter of the mold is 22.8 inches, how many cubic inches of plastic will it take to fill the mold? Use 3.14 for \( \pi \).

To find the volume of the spherical mold, we can use the formula for the volume of a sphere: \[ V = \frac{4}{3} \pi r^3 \] Given that the diameter of the mold is 22.8 inches, we can find the radius by dividing the diameter by 2: \[ r = \frac{22.8}{2} = 11.4 \text{ inches} \] Now, we can substitute the value of the radius into the formula for the volume of the sphere: \[ V = \frac{4}{3} \times 3.14 \times (11.4)^3 \] Let's calculate the volume of the plastic needed to fill the mold. Calculate the value by following steps: - step0: Calculate: \(\frac{4}{3}\times 3.14\times 11.4^{3}\) - step1: Convert the expressions: \(\frac{4}{3}\times 3.14\left(\frac{57}{5}\right)^{3}\) - step2: Multiply the terms: \(\frac{314}{75}\left(\frac{57}{5}\right)^{3}\) - step3: Evaluate the power: \(\frac{314}{75}\times \frac{57^{3}}{5^{3}}\) - step4: Rewrite the expression: \(\frac{314}{75}\times \frac{3^{3}\times 19^{3}}{5^{3}}\) - step5: Rewrite the expression: \(\frac{314}{3\times 25}\times \frac{3^{3}\times 19^{3}}{5^{3}}\) - step6: Reduce the numbers: \(\frac{314}{25}\times \frac{3^{2}\times 19^{3}}{5^{3}}\) - step7: Multiply the fractions: \(\frac{314\times 61731}{25\times 5^{3}}\) - step8: Multiply: \(\frac{19383534}{5^{5}}\) - step9: Evaluate the power: \(\frac{19383534}{3125}\) The volume of the plastic needed to fill the mold is approximately 6202.73 cubic inches.