Question

A solid is of the cylindrical shape and its both ends are hemispherical. The total height of the solid is 19 cm and the radius of the cylinder is 3.5 cm. Find the volume and total surface area of the solid.

Hint:

For composite solids, always break down the problem into its basic geometric shapes. Calculate the dimensions of each part first before applying the formulas. For surface area, only include the surfaces that are exposed on the outside.

Answer 1

Step 1: Understanding the Concept:
The solid is a composite shape made of a central cylinder and two hemispheres at its ends. To find the total volume and total surface area, we need to calculate these properties for each component and then sum them up.
Step 2: Key Formula or Approach:
Let \(r\) be the radius and \(h\) be the height of the cylindrical part.
1. Volume of Cylinder = \( \pi r^2 h \)
2. Volume of two Hemispheres (one Sphere) = \( \frac{4}{3} \pi r^3 \)
3. Total Volume = Volume of Cylinder + Volume of two Hemispheres
4. Curved Surface Area (CSA) of Cylinder = \( 2\pi r h \)
5. CSA of two Hemispheres (Surface Area of one Sphere) = \( 4\pi r^2 \)
6. Total Surface Area (TSA) of Solid = CSA of Cylinder + CSA of two Hemispheres
Step 3: Detailed Explanation:
Dimensions:
Radius of cylinder and hemispheres, \( r = 3.5 \text{ cm} = \frac{7}{2} \text{ cm} \).
The height of the two hemispherical ends is equal to the diameter, \( 2r = 2 \times 3.5 = 7 \text{ cm} \).
Height of the cylindrical part, \(h\) = Total height - Height of two hemispheres
\[ h = 19 - 7 = 12 \text{ cm} \] Volume Calculation:
Total Volume = \( \pi r^2 h + \frac{4}{3} \pi r^3 = \pi r^2 \left( h + \frac{4}{3}r \right) \)
\[ V = \frac{22}{7} \times \left(\frac{7}{2}\right)^2 \left( 12 + \frac{4}{3} \times \frac{7}{2} \right) \] \[ V = \frac{22}{7} \times \frac{49}{4} \left( 12 + \frac{14}{3} \right) \] \[ V = \frac{11 \times 7}{2} \left( \frac{36+14}{3} \right) = \frac{77}{2} \left( \frac{50}{3} \right) \] \[ V = \frac{77 \times 25}{3} = \frac{1925}{3} \approx 641.67 \text{ cm}^3 \] Total Surface Area Calculation:
TSA = CSA of Cylinder + CSA of two Hemispheres
TSA = \( 2\pi r h + 4\pi r^2 = 2\pi r (h + 2r) \)
\[ \text{TSA} = 2 \times \frac{22}{7} \times \frac{7}{2} \left( 12 + 2 \times \frac{7}{2} \right) \] \[ \text{TSA} = 22 (12 + 7) \] \[ \text{TSA} = 22 \times 19 = 418 \text{ cm}^2 \] Step 4: Final Answer:
The volume of the solid is approximately 641.67 cm\(^3\) and the total surface area is 418 cm\(^2\).