A piece of paper is a semi circular region of radius 10cm.It is rolled to form right circular cone.the slant height and height of the cone are?

Question

Answer

Diameter of the semicircular paper = circumference of the base of the cone
=> 20 = 2πr
=> r = 10/π cm

Radius of the semi circle = height of the cone
=> h = 10 cm

If slant height of the cone is l then,
l² = h² + r²
On substituting the values of r and h,
l = 10.49 cm

Answer

Solution:

Radius of the semi-circular paper:

Given, a semi-circular paper of radius 10cm.

Therefore, the radius of the paper(r) = 10cm

Forming the cone:

When the paper is rolled to form a cone, the circumference of the base of the cone is equal to the circumference of the semi-circular paper.

So, the circumference of the base of the cone = 2πr = 2 x π x 10 = 20π cm

The slant height of the cone is the length of the paper's radius, which is the height of the cone.

Calculating the height of the cone:

Let h be the height of the cone.

Then, we can use the Pythagorean theorem to find the slant height of the cone.

Slant height of the cone=√(r^2+h^2)

As r=10cm, we can substitute it in the above equation.

Slant height of the cone=√(10^2+h^2)

Also, the circumference of the base of the cone = 20π cm

So, the radius of the cone(r')= (20π)/(2 x π)=10cm

Therefore, using the formula for the surface area of a cone, we have:

πr'(r'+slant height of cone)=π x 10(10+√(10^2+h^2))=20π²+10π√(10^2+h^2)

The area of the semi-circular paper is half the area of the circle of radius 10, so it is:

1/2 x πr²=1/2 x π x 10²=50π

Equating both the areas, we get:

20π²+10π√(10^2+h^2)=50π

Simplifying the above equation, we get:

2π²+π√(10^2+h^2)=5π

π²+π√(10^2+h^2)/2.5=1

π√(10^2+h^2)/2.5=1-π²

π√(10^2+h^2)=2.5(1-π²)

√(10^2+h^2)=(2.5(1-π²))/π

10^2+h^2=((2.5(1-π²))/π)²

h^2=((2.5(1-π²))/π)²-10²

h=√(((2.5(1-π²))/π)²-10²)

Final Answer:

Slant height of the cone=

Answer

Radius of the cone is equal to the height of the cone
l^2=r^2+h^2
l^2=100+100
=200

Similar Class 10 Doubts

A right circular cylinder and a cone have equal base and equal height if they curved surface area are in the ratio 8:5 show that the ratio between radius of their base to their height is 3:4?

A right circular cone of radius 4 cm and slant height 5 cm is curved out from a cylindrical piece of wood of same radius and height 5 cm. The surface area of the remaining wood is

A rocket is in the form of a circular cylinders closed at the lower end of cone of same radius is attach to a top .the radius of cone is 2.5 and height 21 m and slant heights of cone 8m . Calculate total surface area of rocket?

Top Courses for Class 10View all

Science Class 10

Social Studies (SST) Class 10

Mathematics (Maths) Class 10

English Grammar Advanced

English Grammar Basic

Top Courses for Class 10

Top Courses for Class 10

Suggested Free Tests

Related Content

About Class 10

Class 10 Study Material

Class 10 All Subjects

NCERT Textbook Solutions

Other Classes

Important Links

Company