Question

The height of a solid right circular cone is 20cm. and its slant height is 25cm. If the height of
a solid right circular cylinder, having as much volume as that of the cone, is 15 cm., then let
us calculate the base diameter of the cylinder.
​

Answer 1

Answer:

The wording of this question is such that I assume the cylinder has ht 15 cm so quickly is what radius of cylinder of ht 15cm has Sam volume of circular based cone of height 20cm and slant heigh 25cm.

The cone has radius 15 cm the ht, slant height and radius make a pythagorian triple 15,20,25 (five times the size of a 345 right triangle)

Thus the volume of this cone is (1/3) Base area * ht so (1/3)*Pi*(15^2)*20 = 1500Pi

Cylinder has volume Base area * ht = pi*(r^2)*15 if this is equal to the cone then

1500Pi = 15(r^2)Pi

So r^2=100

r=10cm

Answer 2

Step-by-step explanation:

$\bf\underline{\underline{\red{Given:-}}}$

• Height of the right circular cone = 20cm

• Slant height of the cone = 25cm

• The volume of a right circular cylinder is equal to the volume of the cone.

• Height of the cylinder = 15cm.

$\bf\underline{\underline{\blue{To\:Find:-}}}$

• The base diameter of the cylinder.

$\bf\underline{\underline{\green{Solution:-}}}$

In the right circular cone:-

Height (h) = 20cm

Slant height (s) = 25cm

By applying Pythagoras Theorem

$\rm \implies {s}^{2} = {h}^{2} + {r}^{2}$

$\rm \implies {r}^{2} = {s}^{2} - {h}^{2}$

$\rm \implies {r}^{2} = {25}^{2} - {20}^{2}$

$\rm \implies {r}^{2} = 625 - 400$

$\rm \implies {r}^{2} = 225$

$\rm \implies {r} = \sqrt{225}$

$\rm \implies {r} = 15$

$\rm \therefore Radius \: of \: the \: cone= 15cm$

Now:-

$\boxed{\rm \leadsto Volume \: of \: cone= \frac{1}{3}\pi {r}^{2}h }$

$\rm = \dfrac{1}{3} \times \pi \times 15 \times 15 \times 20$

$\rm = \dfrac{4500\pi}{3}$

$\rm = 1500\pi$

$\underline{ \underline{ \: \: \: \rm Volume \: of \: cone = 1500\pi {cm}^{3} \: \: \: }}$

Now, we need to find the radius of the cylinder

• Height of the cylinder = 15cm

$\boxed{\rm \leadsto Volume \: of \: cylinder= \pi {r}^{2}h }$

Given, Volume of the cone = Volume of the cylinder.

$\rm \implies1500\pi = \pi \times {r}^{2} \times 15$

$\rm \implies \dfrac{1500\pi}{15\pi} = {r}^{2}$

$\rm \implies 100 = {r}^{2}$

$\rm \implies r = \sqrt{100}$

$\rm \implies r = 10cm$

$\rm Radius \: of \: cylinder= 10cm$

$\rm Diameter\: of \: cylinder= 2(Radius)$

$\rm= 2(10)$

$\rm= 20$

$\underline{\boxed{\purple{\rm \therefore Base\: diameter\: of \: cylinder = 20cm}}}$