Question

») What is the volume of a ball with adiameter of 6 centimeters? Use 3.14for (pie)How is the volume of a sphere related tothe volume of a cone that has the samecircular base and height?

Answer 1

Volume of a sphere and a cone

We have that the equation of the volume of a sphere is given by:

`V=(4)/(3)\pi r^3`

We have that the radius of a sphere is half the diameter of it:

Then, the radius of this sphere is

r = 6cm/2 = 3cm

Finding the volume of a sphere

We replace r by 3 in the equation:

`\begin{gathered} V=(4)/(3)\pi r^3 \\ \downarrow \\ V=(4)/(3)\pi\cdot3^3 \end{gathered}`

Since 3³ = 3 · 3 · 3 = 27

`\begin{gathered} V=(4)/(3)\pi3^3 \\ \downarrow \\ V=(4)/(3)\pi\cdot27 \end{gathered}`

If we use π = 3.14:

`\begin{gathered} V=(4)/(3)\pi\cdot27 \\ \downarrow \\ V=(4)/(3)\cdot3.14\cdot27 \\ \downarrow \\ V=4.18\bar{6}\cdot27 \end{gathered}`

Rounding the first factor to the nearest hundredth (two digits after the decimal), we have:

4.18666... ≅ 4.19

Then, we have that:

`\begin{gathered} V=(4)/(3)\pi\cdot27 \\ \downarrow \\ V=4.19\cdot27 \\ =113.13 \end{gathered}`

Then, we have that:

Finding the volume of a cone

We have that the volume of a cone is given by:

`V=(1)/(3)\pi r^2h`

where r is the radius of its base and h is the height:

Then, in this case

r = 3

h = 6

and

π = 3.14

Replacing in the equation for the volume:

`\begin{gathered} V=(1)/(3)\pi r^2h \\ \downarrow \\ V=(1)/(3)\cdot3.14\cdot3^2\cdot6 \end{gathered}`

Then, we have:

3² = 9

`\begin{gathered} V=(1)/(3)\cdot3.14\cdot3^2\cdot6 \\ \downarrow \\ V=(1)/(3)\cdot3.14\cdot9\cdot6 \\ V=(1)/(3)\cdot3.14\cdot54 \\ V=56.52 \\ \end{gathered}`

Answer: the volume of the cone that has the same circular base and height is 56.52 cm³