Plastic Factory A certain machine in a factory fills a spherical mold with plastic. If the diameter of the mold is 24.2 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 24.2 inches, we can find the radius by dividing the diameter by 2: \[ r = \frac{24.2}{2} = 12.1 \text{ inches} \] Now, we can substitute the value of the radius into the formula for the volume of the sphere: \[ V = \frac{4}{3} \times 3.14 \times (12.1)^3 \] Let's calculate the volume of the plastic needed to fill the mold. Calculate the value by following steps: - step0: Calculate: \(\frac{4}{3}\times 3.14\times 12.1^{3}\) - step1: Convert the expressions: \(\frac{4}{3}\times 3.14\left(\frac{121}{10}\right)^{3}\) - step2: Multiply the terms: \(\frac{314}{75}\left(\frac{121}{10}\right)^{3}\) - step3: Evaluate the power: \(\frac{314}{75}\times \frac{121^{3}}{10^{3}}\) - step4: Rewrite the expression: \(\frac{2\times 157}{75}\times \frac{121^{3}}{10^{3}}\) - step5: Rewrite the expression: \(\frac{2\times 157}{75}\times \frac{121^{3}}{2^{3}\times 5^{3}}\) - step6: Reduce the numbers: \(\frac{157}{75}\times \frac{121^{3}}{2^{2}\times 5^{3}}\) - step7: Multiply the fractions: \(\frac{157\times 121^{3}}{75\times 500}\) - step8: Multiply: \(\frac{157\times 121^{3}}{37500}\) The volume of the plastic needed to fill the mold is approximately 7416.935 cubic inches.