A Toy consists of a base that is the section of a sphere and a conical top. The volume of the conical top is 30 π sq. units and its height is 10 units. The total height of the toy is 19 unit. Then the volume of the sphere (in cubic units) is.a)500π/3b)128π/3c)216π/3d)Can’t be determinedCorrect answer is option 'A'. Can you explain this answer?

Question

A Toy consists of a base that is the section of a sphere and a conical top. The volume of the conical top is 30 &pi; sq. units and its height is 10 units. The total height of the toy is 19 unit. Then the volume of the sphere (in cubic units) is.a)500&pi;/3b)128&pi;/3c)216&pi;/3d)Can&rsquo;t be determinedCorrect answer is option 'A'. Can you explain this answer?

Answer

The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.

Let's assume the radius of the base of the conical top is r, and the height of the conical top is h. Since the volume of the conical top is 30, we can write:

30 = (1/3)πr^2h

Now, let's consider the base of the toy, which is the section of a sphere. The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere.

Since the base of the toy is a section of a sphere, we need to find the radius of the sphere that corresponds to the base. Let's call this radius R.

Now, let's find the relationship between the radius of the conical top (r) and the radius of the sphere (R).

The base of the toy is a section of a sphere, so the volume of the base can be written as a fraction of the volume of the whole sphere. The fraction can be calculated by dividing the volume of the base by the volume of the whole sphere.

The volume of the base is given by (2/3)πR^3, where R is the radius of the sphere.

So, the fraction of the volume of the base to the volume of the whole sphere is:

(2/3)πR^3 / ((4/3)πR^3) = 2/4 = 1/2

Since the volume of the base is half the volume of the whole sphere, we can write:

(2/3)πR^3 = (1/2) * ((4/3)πR^3)

Simplifying this equation gives:

(2/3)πR^3 = (2/3)πR^3

This means that the radius of the conical top (r) is equal to the radius of the sphere (R).

Now, let's substitute the value of r into the equation for the volume of the conical top:

30 = (1/3)πr^2h

30 = (1/3)πR^2h

Let's assume the height of the conical top is H.

30 = (1/3)πR^2H

Now, we have two equations:

30 = (1/3)πR^2H
30 = (1/3)πR^2h

Since the radius of the conical top (r) is equal to the radius of the sphere (R), we can write:

H = h

Now, we have two equations:

30 = (1/3)πR^2H
30 = (1/3)πR^2H

Since the left-hand side and right-hand side of the two equations are equal, we can conclude that the height of the conical top (H) is equal to the height of the base (h).

Therefore, the volume of the conical top is 30 cubic units and the heights of both the conical top and the base are equal.

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