Question

A cone has a volume V, a radius r, and a height of 12 cm. a.) A cone has the same height and of the radius of the original cone. Write an expression for it's volume. b.) A cone has the same height and 3 times the radius of the original cone. Write an expression for its volume.

Answer 1

expressionGiven from the original,

Volume=V; radius= r and height = 12cm

Please note that the volume V, of a cone, is calculated with the formula

`\begin{gathered} The\text{Volume of a Cone= }(1)/(3)* Circular\text{ Base Area}*\text{Height} \\ =(1)/(3)*\Pi r^2* h \end{gathered}`

For question (a)

height is the same as the original but the radius is

`\text{new radius,R= }(1)/(3)of\text{ r=}(1)/(3)* r=(r)/(3)`

The expression for the new volume will give us

`\begin{gathered} \text{Note, V}_(cone)=(1)/(3)*\Pi R^2* H \\ H=h=12;\text{ R=}(r)/(3) \\ The\text{ New Volume=}(1)/(3)*\Pi*((r)/(3))^2*12 \\ =(1)/(3)*3.14*(r^2)/(9)*12 \\ =(37.68r^2h)/(27) \\ =1.396r^2cm^3 \end{gathered}`

(a) Therefore, The expression for the new volume is 1.396r²cm³

(b) At this time, the height has the height as the original cone but 3 times the radius of the original cone.

Note that the new height still remains as h, while the new radius will be

`\begin{gathered} \text{New height= h} \\ Then\text{ew radius= 3}* r=3r \end{gathered}`

Substituting the new height and the new radius into the formula for the volume of the cone will give

`\begin{gathered} V_(cone)=(1)/(3)*\Pi* r^2* h \\ \text{the new radius =3r and the new height is still h} \\ V_(cone)=(1)/(3)*\Pi*(3r)^2*12 \\ =(1)/(3)*3.14*9r^2*12 \\ =(3.14*9r^2*12)/(3) \\ =113.04r^2cm^3 \end{gathered}`

(b) Therefore, the expression for the new volume is 113.04r²cm³