Name: Convert Degree Into Radian & Radian Into Degree
Uploaded: 2019-07-19
Duration: 7 min
Description: Let’s understand the concept of circular and radian measure of an angle. We will also learn how to convert degrees to radians and vice versa with the help of...

Question

The diagram shows the cross-section of two cylindrical metal rods of radii $x cm$ and $y cm$ . A thin band, of length $P cm$ , holds the two rods tightly together. Show that $P = 4 \sqrt{x y} + π (x + y) + 2 (x - y) \sin^{- 1} (\frac{x - y}{x + y})$ .

Accepted Answer

Given that, The wire is made up of two arcs of circles joined by two straight lines. The straight part of the wire runs along two lines that are tangent to both circles. Drawing in the radii of both circles and the straight line joining we have

Let the length of the wire be $y cm$

From triangle $A B C$

Using Pythagoras theorem

$a = \sqrt{(x + y)^{2} - (x - y)^{2}}$

$a = \sqrt{x^{2} + 2 x y + y^{2} - x^{2} + 2 x y - y^{2}}$

$a = \sqrt{4 x y} or a = 2 \sqrt{x y}$

Using trigonometry

$\sin θ = \frac{x - y}{x + y}$

$θ = \sin^{- 1} (\frac{x - y}{x + y})$

To find the length of the arcs,

The reflex angle at $x cm$ is $2 π - (\frac{π}{2} - θ) - (\frac{π}{2} - θ)$

$= 2 π - \frac{π}{2} + θ - \frac{π}{2} + θ$

$= π + 2 θ$

The obtuse angle at $B$ is $2 π - \frac{π}{2} - \frac{π}{2} - θ - θ$

$= π - 2 θ$

Arc length $= r θ$

Length of the left arc is $x (π + 2 θ)$

Length of the right arc is $y (π - 2 θ)$

Total length of the belt is $P = a + a + x (π + 2 θ) + y (π - 2 θ)$

$= 2 \sqrt{x y} + 2 \sqrt{x y} + x π + y π + 2 x θ - 2 y θ$

$= 4 \sqrt{x y} + π (x + y) + 2 θ (x - y)$

$P = 4 \sqrt{x y} + π (x + y) + 2 (x - y) \sin^{- 1} (\frac{x - y}{x + y})$ , since $θ = \sin^{- 1} (\frac{x - y}{x + y})$ .

Hence proved.

The diagram shows the cross-section of two cylindrical metal rods of radii $x cm$ and $y cm$ . A thin band, of length $P cm$ , holds the two rods tightly together. Show that $P = 4 \sqrt{x y} + π (x + y) + 2 (x - y) \sin^{- 1} (\frac{x - y}{x + y})$ .

Important Questions on Circular Measure

Learn and Prepare for any exam you want !

The diagram shows the cross-section of two cylindrical metal rods of radiix cm and y cm. A thin band, of length P cm, holds the two rods tightly together. Show that P=4xy+π(x+y)+2(x-y)sin-1x-yx+y.

Important Questions on Circular Measure

Learn and Prepare for any exam you want !

The diagram shows the cross-section of two cylindrical metal rods of radii $x cm$ and $y cm$ . A thin band, of length $P cm$ , holds the two rods tightly together. Show that $P = 4 \sqrt{x y} + π (x + y) + 2 (x - y) \sin^{- 1} (\frac{x - y}{x + y})$ .