Question

A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use \(\pi = \frac{22}{7}\))

Hint:

Always group your \(\pi r^2\) terms before calculating. It saves you from multiplying by \(22/7\) multiple times and reduces rounding errors.

Answer 1

Step 1: Understanding the Concept:
The solid consists of one cylinder and two hemispheres (which together make one full sphere). We need to subtract the radii of the hemispheres from the total height to find the height of the cylindrical part.
Step 2: Key Formula or Approach:
1. Volume of Cylinder = \(\pi r^2 h\) 2. Volume of Sphere = \(\frac{4}{3} \pi r^3\)
Step 3: Detailed Explanation:
1. Radius \(r = 7/2 = 3.5\) cm. 2. Height of cylinder \(h = \text{Total height} - 2r = 20 - 7 = 13\) cm. 3. Total Volume = Volume of Cylinder + Volume of Sphere: \[ V = \pi r^2 h + \frac{4}{3} \pi r^3 = \pi r^2 \left( h + \frac{4}{3}r \right) \] 4. Substitute values: \[ V = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times \left( 13 + \frac{4}{3} \times \frac{7}{2} \right) \] \[ V = \frac{77}{2} \times \left( 13 + \frac{14}{3} \right) = 38.5 \times \left( \frac{39 + 14}{3} \right) \] \[ V = 38.5 \times \frac{53}{3} \approx 680.17 \text{ cm}^3 \]
Step 4: Final Answer:
The total volume of the solid is approximately 680.17 cm\(^3\).