Question

Find the volume of a solid in the form of a right circular cylinder with hemispherical ends whose length is 2.7m and the diameter of each hemispherical end is 0.7m.

Answer 1

Hint: We can split the solid into a cylinder and 2 hemispheres. The radius of the hemisphere will be the radius of the cylinder. The height of the cylinder can be obtained by subtracting the radius of the hemisphere from both ends. Then we can find the volume of the cylinder using its height and radius and the volume of the 2 hemispheres using the radius. Finally, add the volumes of the cylinder and the hemispheres to get the required solution.

Complete step by step answer:

We can draw a figure to represent the given solid.

We can split the given solid as a cylinder and 2 hemispheres. It is given that the height of the solid is 2.7m and the diameter of each hemispherical end is 0.7m.
Let H be the height of the solid and h be the height of the cylinder. Let r be the radius of the hemisphere. According to the question we have,
$H = 2.7m$
Then the radius of the hemisphere will be half of the diameter.
$ \Rightarrow r = \dfrac{{0.7}}{2}m$
$ \Rightarrow r = 0.35m$
From the figure, we can see that the radius of the cylinder is the same as the radius of the hemisphere.
From the figure, we can understand that the height of the cylinder is the height of the solid minus the radius of the hemisphere from both ends.
$ \Rightarrow h = H - r - r$
On substituting the value of H and r, we get,
$ \Rightarrow h = 2.7 - 0.35 - 0.35$
On further calculation we get,
$ \Rightarrow h = 2.0m$
Now we can find the volume of the cylinder. It is given by the equation,
$ \Rightarrow {V_c} = \pi {r^2}h$
On substituting the values, we get,
$ \Rightarrow {V_c} = \dfrac{{22}}{7} \times {\left( {0.35} \right)^2} \times 2$
On simplifying we get,
$ \Rightarrow {V_c} = 22 \times 0.35 \times 0.05 \times 2$
After multiplication, we get,
$ \Rightarrow {V_c} = 0.77{m^3}$
Now we can find the volume of the hemisphere. It is given by the equation,
$ \Rightarrow {V_h} = \dfrac{2}{3}\pi {r^3}$
On substituting the values, we get,
$ \Rightarrow {V_h} = \dfrac{2}{3} \times \dfrac{{22}}{7} \times {\left( {0.35} \right)^3}$
On simplification we get,
$ \Rightarrow {V_h} = \dfrac{2}{3} \times 22 \times 0.35 \times 0.35 \times 0.05$
After multiplication, we get,
$ \Rightarrow {V_h} = 0.0898{m^3}$
Now the volume of the solid is given by adding the volume of the cylinder with 2 times the volume of the hemisphere.
$ \Rightarrow V = {V_c} + 2 \times {V_h}$
On substituting the values, we get,
$ \Rightarrow V = 0.77 + 2 \times 0.0898$
$ \Rightarrow V = 0.77 + 0.1796$
$ \Rightarrow V = 0.9496{m^3}$
Therefore, the volume of the body is $0.9496{m^3}$.

Note: Volume of a body is the amount of space occupied by it in a three-dimensional space. Here we split the body into 3 parts whose volumes can be found using the equations we know. We must draw a figure for a better understanding of the question. We may make errors while taking the volume of the hemisphere. We must make sure that we add twice the volume of the hemisphere as both ends have a hemisphere. We must take care while doing decimal multiplications. We must find the value of the decimal up to 3 decimal places at least.