Question

the base of a solid is the region enclosed by a circle having a radius of 7 cm. find the volume of the solid if all plane sections perpendicular to a fixed diameter of the base are equilateral triangles

Answer 1

Answer:

$V=621.8\ cm^3$

Step-by-step explanation:

we know that

If the base of the solid is a circle and all plane sections perpendicular to a fixed diameter of the base are equilateral triangles

then

The solid is a cone

The volume of a cone is equal to

$V=\frac{1}{3}\pi r^{2}h$

we have

$r=7\ cm$

Find out the height of the cone

Remember that

If all plane sections perpendicular to a fixed diameter of the base are equilateral triangles

then

The length side of each equilateral triangle is equal to the diameter of the base

Applying the Pythagoras Theorem find out the height of the cone

$D^{2}=r^{2} +h^{2}$

we have

$D=(2)7=14\ cm$ ----> the diameter is two times the radius

$r=7\ cm$

substitute and solve for h

$14^{2}=7^{2} +h^{2}$

$h^{2}=14^{2}-7^{2}$

$h^{2}=147$

$h=\sqrt{147}\ cm$

Find the volume

$V=\frac{1}{3}\pi (7)^{2}\sqrt{147}$

$V=198.03\pi\ cm^3$

assume

$\pi =3.14$

$V=198.03(3.14)=621.8\ cm^3$

Answer 2

The volume of the solid if all plane sections perpendicular to a fixed diameter of the base are equilateral triangles is 622.13 cm³.

As we know the base of the solid is a circle of radius 7cm and plane sections perpendicular to a fixed diameter of the base are equilateral triangles. So, This solid is nothing but a cone with a circular base with a radius of 7cm.

What is an equilateral triangle?

An equilateral triangle is a triangle having all sides and all angles equal to each other.

Since plane sections are equilateral triangles,

This means the slant height of the cone will be a side of the equilateral triangle as well as side length will be equal to the diameter of the base.

So, slant height h = 14cm

radius r = 7cm

so height = $\sqrt{14^{2}-7^{2} }$ = $7\sqrt{3}$

So, the volume of the cone = $\frac{1}{3} \pi r^{2} h$

The volume of the cone = $\frac{1}{3}* \pi *7^{2}* 7\sqrt{3}$

The volume of the cone = 622.13 cm³.

Therefore, the volume of the solid if all plane sections perpendicular to a fixed diameter of the base are equilateral triangles is 622.13 cm³.

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https://brainly.com/question/1082469