Question

If a cone has a height of 5 cm and a volume of 128π cm³
find the diameter of the circular base.

Answer 1

The volume of a cone with respect to its diameter is:

V=(hπd^2)/12  solving this for d we have:

d=√[(12V)/(hπ)], we are given that V=128π and h=5 so

d=√(1536π)/(5π)

d=√(1536/5)

d=√307.2 cm

d≈17.53 cm (to nearest hundredth)

Answer 2

Using the formula of a volume of the cone we can find its area:
$V=\frac{1}{3}A*h$
$A=\frac{3V}{h}$

Using the formula of an area of the circle we can now find the radius of cone's base:

$A=\pi r^2$
$\frac{3V}{h}=\pi r^2$
$r^2=\frac{3V}{h \pi}$
$r=\sqrt{\frac{3V}{h \pi}$

The diameter of circle is twice of its radius, so:
$d=2\sqrt{\frac{3V}{h \pi}$
$d=2\sqrt{\frac{3*128\pi}{5\pi}}=2\sqrt{\frac{384}{5}}\approx17.53$

So, the diameter of cone is approximately equal to 17.53cm.