A right circular cylinder and a right circular cone have equal radii of their bases
and equal heights. If their curved surfaces are in the ratio 24:13, show that the
radius and the height of each are in the ratio 5:12.
Fig
Answers
Step-by-step explanation:
Given :-
A right circular cylinder and a right circular cone have equal radii of their bases and equal heights. If their curved surfaces are in the ratio 24:13.
To find :-
Show that the radius and the height of each are in the ratio 5:12.
Solution:-
Given that
A right circular cylinder and a right circular cone have equal radii of their bases and equal heights
Let the radius of the cylinder be r units
Then the radius of the cone = r units
Let the radius of the Cylinder be h units
Then the radius of the cone = h units
The Curved Surface Area of the Cylinder
= 2πrh sq.units
The Curved Surface Area of the cone
= πrl sq.units
We know that
Slant height of the cone = l=√(r^2+h^2) units
So, Curved Surface Area of the cone
= πr(√(r^2+h^2)) sq.units
Now
The ratio of the Curved Surface Areas of the Cylinder to the Cone
= 2πrh : πr(√(r^2+h^2))
= 2h : √(r^2+h^2) ---------(1)
Given that
Their curved surfaces ares in the ratio 24:13.
=> 2h : √(r^2+h^2)) = 24:13
=> 2h / √(r^2+h^2)) = 24 / 13
On applying cross multiplication then
=> 2h×13 = √(r^2+h^2))×24
=> 26 h = √(r^2+h^2))×24
=> 26/24 = (r^2+h^2) / h
=> 13/12 = √(r^2+h^2) / h
On squaring both sides then
=> (13/12)^2 = [√(r^2+h^2) / h]^2
=> 169/144 = (r^2+h^2)/h^2
=> 169/144 = (r^2/h^2)+(h^2/h^2)
=> 169 / 144 = (r^2/h^2)+1
=> (169/144)-1 = r^2/h^2
=>(169-144)/144 = r^2/h^2
=> 25/144 = r^2/h^2
=> r^2/h^2 = 25/144
=> (r/h)^2 = 25/144
=>r/h = √(25/144)
=> r/h = 5/12
=> r:h = 5:12
Hence , Proved.
Answer:-
The ratio of the radius of the base and height of each solids is 5:12
Used formulae:-
- Curved Surface Area of the right circular cylinder = 2πrh sq.units
- Curved Surface Area of the right circular cone = πrl sq.units
- Slant height of the cone = √(h^2+r^2) units
- r = radius
- h = height
- π=22/7
- a:b can be written as a/b