A person observed that he required 30 s time to cross a circular ground along its diameter than to cover it once along the boundary. If his speed was 30m/min, than the radius of the circular ground is (take ? = 22/7)

The man takes 3600 s for 14.4 km
The man will take 88 s for
14.4 x (88/3600) = 352/1000 km = 352 m

Now, circumference of circular field = 352 m
? 2?r = 352 m
2 x (22/7) x r = 352
? r = 56 m

Therefore, area of the field = ?r²
= (22/7) x 56 x 56
= 8 x 22 x 56 m²
= 9856 sq m.

Length of to the rope = Radius of circle
According to the question,
?r² = 154
? r² = 154 x (7/22) = 7 x 7 = 49
? r = ?49 = 7 m

Let original radius be r.
Then, according to the questions,
? (r + 1)² - ?r² = 22
? ? x [(r + 1)² - r²] = 22
? (22/7) x (r + 1 + r ) x (r + 1 - r) = 22
? 2r + 1 = 7
? 2r = 6
? r = 6/2 = 3 cm

Increase in circumference of circle = 5%
? Increase in radius is also 5%.
Now, increase in area of circle = 2a + (a²/100) %
Where, a = increase in radius= 2 x 5 + (5 x 5)/100 % = 10.25%

Ratio of the areas of the circumcircle and incircle of a square
= [(Diagonal)²?] / [(Side)²?]
= [(Side x ?2)²] / (Side)² = 2/1 or 2 : 1

Length ot rectangle = 40 m
Let breadth of = k
Then, according of the question,
(40 + k)15 = 40k
? 600 + 15k = 40k
? 25k = 600
? k = 24 m

Let the length of rectangle = L m
? Breadth of rectangle = B m
Using conditions from the question,
L - B = 23 ....(i)
2(L + B) = 206
L + B = 103 ....(ii)

On adding Eqs. (i) and (ii), we get
2L = 126
? L = 63 m
? B = 103 - 63 = 40 m

Then , area of rectangle = L x B
= 63 x 40
= 2520 m² .

Let the width of the rectangle = k units
? Length = (2k + 5) units.

According to the question,
Area = k(2k + 5)
? 75 = 2k² + 5k
? 2k² + 5k - 75 = 0
? 2k² + 15k - 10k - 75 = 0
? k(2k + 15) - 5(2k + 15) = 0
? (2k + 15) (k - 5) = 0
? k = 5 and -15/2
Width cannot be negative.
? Width = 5 units
? Length = 2x + 5 = 2 x 5 + 5 = 15 unit

? perimeter of the rectangle = 2(15 + 5)
= 40 units

Original breadth of rectangle = 720/30 = 24 cm
Now , area of rectangle = (5/4) x 720 = 900 cm²
? New length of rectangle = 900/24 = 37.5 cm
? New perimeter of rectangle = 2(l+ b)
= 2(37.5 + 24 )
= 2 x 61.5
= 123 cm

Let the sides of trapezium be 5k and 3k, respectively

According to the question,
(1/2) x [(5k + 3k) x 12] = 384
? 8k = (384 x 2)/12 = 64
? k = 64/8 = 8 cm

Length of smaller of the parallel sides = 8 x 3 = 24 cm