A solid toy is in the form of a hemisphere surmounted by a right circular cone.The height of the cone is 4cm.The diameter of the base is 8cm.Determine the volume of the toy if the cube circumscribes the toy then find the difference of the cube and toy .Also find the total surface area of the toy?
Answers
1408/7cm3
and 2176/7 cm3
and total surface area= 352(3+√2)/7
Given:
The height of the cone = 4cm
The diameter of the base = 8cm
To Find:
The volume and TSA of the toy.
The difference between the cube and toy
Solution:
The height of the cone is given as 4cm.
The radius of the cone is equal to the radius of the hemisphere as the toy is in form of a hemisphere surmounted by the right circular cone.
So, the radius of cone = radius of the hemisphere = 4cm
Now, to find the Volume of the toy we need to find the volume of the cone and the volume of the hemisphere and add them.
So, the formula to find the volume of the cone is 1/3πr²h, and the formula for the volume of a hemisphere is 2/3πr³ where 'r' is the radius, 'h' is the height, and π = 22/7
Then,
The volume of toy = volume of the cone + volume of a hemisphere
= 1/3πr²h + 2/3πr³
[Taking πr² common from the formula]
= πr²[1/3h + 2/3r]
We now substitute the values,
= 22/7×4×4[1/3×4+2/3×4]
= 22/7×16[4/3+8/3]
= 22/7×16[12/3]
= 22/7 × 16 × 4
= 138.16cm³
Therefore, the volume of the toy = 138.16cm³
Now, the dimension of the cube is given as 8cm
Volume of the cube = side³
= 8³
= 512cm³
Hence, volume of the cube = 512cm³
The difference to the toy and cube = volume of the cube - volume of the toy
= 512 - 138.16
= 373.84cm³
Therefore, the difference between the cube and the toy = 373.84cm³
Now, we find the total surface area of the toy.
The formula for the total surface area of the cone is CSA of the cone + CSA of the hemisphere. = πrl + 2πr²
Noe, we need to find 'l' which is the slant height of the cone.
The slant height of the cone using Pythagoras theorem,
l = √h²+r²
= √(4)² + (4)²
= √16 + 16
= √32
=4√2cm
∴ slant height 'l' = 4√2cm
Then, the total height of the cone = CSA of the cone + CSA of the hemisphere
= πrl + 2πr²
= πr[l + 2r]
= 22/7×4[4√2+2(4)]
= 22/7×4×4(√2+2)
= 352/7 (√2+2)
= 5.2(√2+2)cm²
Therefore, the total surface of the area = 5.2(√2+2)cm²