A Toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere .If the radius of base of the cone is 21 cm and its volume is 2/3 of the volume of hemisphere calculate the height of cine and surface area of the toy.

The missing part is the height of the cone, but it is 2/3 the volume of the hemisphere (semi-sphere).

volume of hemisphere = (1/2 (4/3)π (21^3) = 6174π
but the volume of the cone is 2/3 of that or 4116π

vol of cone = (1/3)π r^2 h
4116π = (1/3)π(21^2)h
12348 = 441h
h = 28

Now use your formulas for surface area of a sphere, take half of that , and use your formula for the surface area of a cone with radius 21 and height 28
Make sure NOT to include the circular base joining them when finding surface area

Volume of cone =

3
2

Volume of hemisphere

3
1

πr
2
h=
3
2

(
3
2

πr
3
) ; r =Common radius h = Cone height

h=
3
4

r=
3
4

(21)=28cm

Slant height, l=
r
2
+h
2


=
21
2
+28
2


=35cm

Total surface area = CSA
cone

+CSA
hemisphere



=πrl+2πr
2
=πr(l+2r)=5082cm
2


Solve any question of Surface Areas and Volumes with:-

To solve this problem, we need to calculate the height of the cone and the surface area of the toy.

Let's start with the volume of the cone.

Given:
Radius of the base of the cone (r) = 21 cm
Volume of the cone = 2/3 * Volume of the hemisphere

The volume of the cone is given by the formula:
Volume of cone = (1/3) * π * r^2 * h

We can rewrite the volume of the cone equation as:
(1/3) * π * 21^2 * h = (2/3) * Volume of the hemisphere

Since the radius of the hemisphere is the same as the base radius of the cone, we can denote it as 'r'. Therefore, the radius of the hemisphere (r) is also equal to 21 cm.

The volume of the hemisphere is given by the formula:
Volume of hemisphere = (2/3) * π * r^3

We can substitute the values into the volume equation of the cone:
(1/3) * π * 21^2 * h = (2/3) * (2/3) * π * 21^3

Simplifying the equation:
21^2 * h = (2/3) * (2/3) * 21^3

Now, let's solve for the height of the cone (h):
h = ((2/3) * (2/3) * 21^3) / (21^2)
= (4/9) * 21
= 84/3
= 28 cm

Therefore, the height of the cone is 28 cm.

Now let's calculate the surface area of the toy.

Surface area of the toy = Surface area of the hemisphere + Surface area of the cone

The surface area of the hemisphere is given by the formula:
Surface area of hemisphere = 2 * π * r^2

Substituting the value of r (21 cm),
Surface area of hemisphere = 2 * π * 21^2

The surface area of the cone is given by the formula:
Surface area of cone = π * r * l, where l is the slant height

To find the slant height (l), we can use the Pythagorean theorem:
l = sqrt(r^2 + h^2)

Substituting the values of r (21 cm) and h (28 cm),
l = sqrt(21^2 + 28^2)
= sqrt(441 + 784)
= sqrt(1225)
= 35 cm

Now, we can calculate the surface area of the cone:
Surface area of cone = π * 21 * 35

Finally, we can calculate the surface area of the toy:
Surface area of the toy = Surface area of hemisphere + Surface area of cone

I hope this helps!

To find the height of the cone and the surface area of the toy, we need to follow these steps:

Step 1: Calculate the volume of the hemisphere.
The formula for the volume of a hemisphere is (2/3) * Pi * r^3, where r is the radius. In this case, the radius of the hemisphere is also the radius of the cone, which is given as 21 cm.
So, the volume of the hemisphere = (2/3) * Pi * (21)^3

Step 2: Calculate the volume of the cone.
The formula for the volume of a cone is (1/3) * Pi * r^2 * h, where r is the radius of the base and h is the height. We need to find the height of the cone, so we'll use the given information that the volume of the cone is 2/3 of the volume of the hemisphere.
(2/3) * (1/3) * Pi * (21)^2 * h = (2/3) * Pi * (21)^3

Step 3: Solve for the height of the cone.
Dividing both sides of the equation by [(2/3) * (1/3) * Pi * (21)^2], we get:
h = [(2/3) * Pi * (21)^3] / [(2/3) * (1/3) * Pi * (21)^2]
Simplifying, we find:
h = 21 cm

Step 4: Calculate the surface area of the toy.
The surface area of the toy consists of three parts: the curved surface area of the hemisphere, the curved surface area of the cone, and the base area of the cone.
The formula for the surface area of a hemisphere is 2 * Pi * r^2, where r is the radius. The formula for the curved surface area of a cone is Pi * r * l, where r is the radius and l is the slant height. The formula for the base area of a cone is Pi * r^2.

Surface area of hemisphere = 2 * Pi * (21)^2
Curved surface area of cone = Pi * (21) * l
Base area of cone = Pi * (21)^2

Step 5: Find the slant height of the cone.
The slant height can be found using the Pythagorean theorem. The slant height (l) and the height of the cone (h) form a right-angled triangle with the radius of the base (21 cm) as the hypotenuse.
Using the formula: l = sqrt(h^2 + r^2), we can calculate the slant height:
l = sqrt((21)^2 + (21)^2)

Step 6: Calculate the surface area of the toy.
Now that we have all the required values, we can calculate the surface area of the toy using the formulas mentioned earlier.

Total surface area of the toy = Surface area of hemisphere + Curved surface area of cone + Base area of cone

I hope this helps!