Question

3. The volume of the cone is 4π dm3 and the diameter of the base is 80 cm. Calculate the cone: (a) the length of the apex; b) the area of the base.

Answer 1

`\begin{gathered} Length\text{ of the apex = 0.75 dm} \\ Area\text{ of the base = 50.24 dm}^2 \end{gathered}`

Step-by-step explanation:

Given:

Volume of the cone = 4π dm³

diameter of the base = 80cm

To find:

a) the length of the apex

b) the base area

A) the apex of a cone is the height of the cone

To get the height, we will apply the formula for the volume of a cone

`\begin{gathered} Volume\text{ of a cone = }(1)/(3)πr²h \\ where\text{ r = radius} \\ h\text{ = height} \end{gathered}`

diameter = 80cm

diameter = 2(radius)

radius = diameter/2 = 80/2 = 40cm

The units are different, so we need to do conversion fromcm to dm

1 dm = 10cm

40cm = 40/10 = 4 dm

substitute the values into the formula:

`\begin{gathered} 4π\text{ = }(1)/(3)*π*4^2* h \\ 3(4π)\text{ = \pi}*16* h \\ h\text{ = }(12π)/(16π) \\ h\text{ = }(3)/(4)dm \\ h\text{ = 0.75 dm} \end{gathered}`

b) The base of a cone is a circle. So, the area of the base will be the area of the circle

`\begin{gathered} Area\text{ of circle = \pi r}^2 \\ let\text{ \pi = 3.14} \\ r\text{ = radius} \end{gathered}`
`\begin{gathered} Area\text{ of the circle = 3.14 }*4^2 \\ \\ Area\text{ of the circle = 50.24 dm} \\ \\ Area\text{ of the base = 50.24 dm}^2 \end{gathered}`