A solid is made from a hemisphere and a cylinder The plane face of the hemisphere coincides with the upper plane face of the cylinder The hemisphere and the cylinder have the same radius. The ratio of the Radius at the cylinder to the height of the cylinder is 1:3 given that the solid has volume 792 pi cm^3 . work out the height of the solid

A solid is made from a hemisphere and a cylinder
The plane face of the hemisphere coincides with the upper plane face
of the cylinder

The hemisphere and the cylinder have the same radius. The ratio of the Radius at the cylinder to the
height of the cylinder is 1:3 given that the solid has volume 792 pi cm^3 . work out the height of the solid

Let's denote the radius of both the hemisphere and the cylinder as "r" and the height of the cylinder as "h".

The volume of the solid is given as 792π cm^3.

The volume of the hemisphere is (2/3)πr^3, and the volume of the cylinder is πr^2h. Since the solid is made up of a hemisphere and a cylinder, we can write the equation:

(2/3)πr^3 + πr^2h = 792π

We can simplify the equation by dividing both sides by π:

(2/3)r^3 + r^2h = 792

Given that the ratio of the radius to the height is 1:3, we can write r = 3h.

Substituting this value into the equation, we get:

(2/3)(3h)^3 + (3h)^2h = 792

Simplifying further:

(2/3)(27h^3) + 9h^3 = 792

18h^3 + 9h^3 = 792

27h^3 = 792

Dividing both sides by 27:

h^3 = 792/27

h^3 = 29.333...

Taking the cube root of both sides:

h ≈ 3.080

Therefore, the height of the solid is approximately 3.080 cm.

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