how do I find the height of a conical shape whose diameter is equal to it's height and the volume is 1m^3

Expert Answers
sciencesolve eNotes educator| Certified Educator

Since the problem provides the value of the volume of conical shape, thus, you need to use the formula that gives the volume of a cone such that:

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

The problem also provides the information that the height of the cone is equal to the diameter of circular base, hence, you should come up with the notation for both dimensions:

`x = h = D`

Since the equation that relates the diameter and the radius is `r = D/2` , hence, you need to substitute `x/2`  for r in equation of volume such that:

`V = (pi*(x/2)^2*x)/3`

`V = pi*x^3/12`

Substituting 1 for V yields:

`1 = pi*x^3/12 =gt 12 = pi*x^3 =gt x^3 = 12/pi =gt x = root(3)(12/pi)`

Hence, since x = h, then the height of the conical shape is `h = root(3)(12/pi) m.`

jeew-m eNotes educator| Certified Educator

The equation for volume of the cone (V) in m^3= (1/3)*pi*(r^2)*h

r = radius of cone in meters

h= height of cone in meters

 

So for our question;

V = 1 m^3

r = h/2      (diameter of cone = height of cone)

 

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

1 = (1/3)*pi*((h/2)^2)*h

1 = (1/3)*pi*((h^2))/4*h

1 = (pi*h^3)/(3*4)

12 = pi*h^3

h^3 = 12/pi

    h = (12/pi)^(1/3)

      = 1.5463 m

 

So the height of the cone is 1.5463 m.