A solid cone of radius r and height h is placed over a solid cylinder having same base radius and height as that of a cone. The total surface area of the combined solid is πr [√(r² + h²) + 3r + 2h]. Is the following statement true or false and justify your answer
Solution:
Given, a solid cone of radius r and height h is placed over a solid cylinder having the same base radius and height as that of cone.
We have to determine if the total surface area of the combined solid is πr [√(r²+h²) + 3r + 2h]
Total surface area of cylinder = curved surface area + area of the bases
Curved surface area is defined as the area of only curved surface leaving the top and bottom bases.
Curved surface area of cylinder = 2πrh
Area of bases = 2πr²
So, total surface area of cylinder = 2πrh + 2πr²
Total surface area of cone = curved surface area + area of base
Curved surface area of cone is defined as the area of only curved surface leaving the bottom base.
Curved surface area of cone = πrl
Where, l is the slant height
Total surface area = πrl + πr²
We know l = √r² + h²
Where, r is the radius of cone
h is the height of cone
So, curved surface area = πr(√r² + h²)
Total surface area of cone = πr(√r² + h²) + πr²
The top circular part of the cylinder and bottom circular part of the cone
The total surface area of the shape so formed = total surface area of cone + total surface area of cylinder - 2(area of the base)
= πrl + πr² + 2πrh + 2πr² - 2πr²
= πrl + πr² + 2πrh
= πr [l + r + 2h]
= πr(√r² + h²) + r + 2h)
Therefore, the total surface area of the shape so formed is πr(√r² + h²) + r + 2h)
✦ Try This: A solid right circular cone of height 120 cm and radius 60 cm is placed in a right circular cylinder full of water of height 180 cm such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is equal to the radius of the cone
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 13
NCERT Exemplar Class 10 Maths Exercise 12.2 Problem 3
A solid cone of radius r and height h is placed over a solid cylinder having same base radius and height as that of a cone. The total surface area of the combined solid is πr [√(r² + h²) + 3r + 2h]. Is the following statement true or false and justify your answer
Summary:
The statement “A solid cone of radius r and height h is placed over a solid cylinder having the same base radius and height as that of a cone. The total surface area of the combined solid is πr [√(r² + h²) + 3r + 2h]” is false
☛ Related Questions:
- A solid ball is exactly fitted inside the cubical box of side a. The volume of the ball is (4/3)πa³. . . . .
- The volume of the frustum of a cone is πh/3[r₁² + r₂² - r₁r₂], where h is vertical height of the fru . . . .
- The capacity of a cylindrical vessel with a hemispherical portion raised upward at the bottom as sho . . . .