Questions: Volume Surface Question 13, 8.4.21 HW Score: 40.52% Points: 0 of 1 Determine the volume of the shaded region. Use the π button on your calculator. The volume of the shaded region is (Simplify your answer. Type an integer or a decimal rounded to the nearest hundredth as needed.)

Solution Steps

The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

Given:

• Diameter of the cylinder = 6 cm, so radius \( r = \frac{6}{2} = 3 \) cm
• Height \( h = 17.4 \) cm

\[ V_{\text{cylinder}} = \pi (3)^2 (17.4) = \pi (9) (17.4) = 156.6\pi \]

The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius.

Given:

• Diameter of the sphere = 5.7 cm, so radius \( r = \frac{5.7}{2} = 2.85 \) cm

\[ V_{\text{sphere}} = \frac{4}{3} \pi (2.85)^3 \approx \frac{4}{3} \pi (23.144625) \approx 30.86\pi \]

There are 3 spheres in the cylinder.

\[ V_{\text{total spheres}} = 3 \times 30.86\pi = 92.58\pi \]

The volume of the shaded region is the volume of the cylinder minus the total volume of the spheres.

\[ V_{\text{shaded}} = V_{\text{cylinder}} - V_{\text{total spheres}} = 156.6\pi - 92.58\pi = 64.02\pi \]

Final Answer