Question

The area of a circular path of uniform width h surrounding a circular region of radius r is
(a)π(2r+h)r
(b)π(2r+h)h
(c)π(h+r)r
(d)π(h+r)h

Answer 1

Answer:

The area of a circular path is  πh(2r + h) .

Among the given options option (b) πh(2r + h) is the correct answer

Step-by-step explanation:

Given :  

Radius of a circular region of a circular path (inner circle) = r  

Uniform width of a circular path = h

Radius of outer circle, R = radius of inner circle + width  

R =  r + h …………(1)

Area of a circular path ,A= Area of outer circle - Area of inner circle

A = πR² - πr²

A = π(R² - r²)

A = π{(r + h) ² - r²}

[From eq 1]

A = π{r² + h² + 2rh - r²}

[(a+ b)² = a² + b² + 2ab ]

A = π{r² - r² + h² + 2rh}

A = π{h² + 2rh}

A = πh(2r + h)

Area of a circular path = πh(2r + h)  

Hence, the area of a circular path is  πh(2r + h) .

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Answer 2

Solution:

Dimensions of the circular path:

Radius of the outer circle = R

Radius of the inner circle = r

width of of the circular path

= h

h = R-r---(1)

R = h+r ----(2)

now,

Area of the circular path (A)

= πR² - πr²

= π(R²-r²)

= π(R+r)(R-r)

/* a²-b²=(a+b)(a-b)*/

= π(h+r+r)h /*from (1)&(2) */

= π(2r+h)h

Therefore,

Option (b) is correct.

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