Question

Both the ends of a metallic solid cylinder are semi-spherical. Its total height is 19 cm and the diameter of the cylinder is 7 cm. Find the weight of the solid if the weight of \(1 \, \text{cm}^3\) of the metal is \(4.5 \, \text{g}\).

Hint:

When the total height includes two hemispheres, always subtract \(2r\) to get the height of the cylindrical part before finding volume.

Answer 1

Step 1: Write the given data.
Total height of the solid \(H = 19 \, \text{cm}\)
Diameter of the cylinder \(= 7 \, \text{cm} \Rightarrow r = 3.5 \, \text{cm}\)
Weight of \(1 \, \text{cm}^3 = 4.5 \, \text{g}\)
Step 2: The solid consists of a cylinder with two hemispherical ends.
Let the height of the cylindrical part be \(h\). Since both ends are hemispheres of radius \(r\), \[ H = h + 2r \] \[ 19 = h + 7 \Rightarrow h = 12 \, \text{cm} \]
Step 3: Write the volume of the solid.
\[ \text{Volume of solid} = \text{Volume of cylinder} + \text{Volume of two hemispheres} \] \[ V = \pi r^2 h + \frac{4}{3}\pi r^3 \]
Step 4: Substitute the values.
\[ V = \pi (3.5)^2 (12) + \frac{4}{3}\pi (3.5)^3 \] \[ V = \pi [147 + \frac{4}{3}(42.875)] = \pi [147 + 57.17] = \pi (204.17) \] \[ V = 3.14 \times 204.17 = 640.9 \, \text{cm}^3 \]
Step 5: Find the weight.
\[ \text{Weight} = \text{Volume} \times \text{Weight per cm}^3 \] \[ = 640.9 \times 4.5 = 2884.05 \, \text{g} \] Step 6: Conclusion.
Hence, the weight of the solid metallic cylinder is \(\boxed{2884.05 \, \text{g} \, \text{or} \, 2.884 \, \text{kg}}\).