Question

there are 24 matchsticks to make a boundary of a park. what will be the largest and shortest area of the park

Answer 1

largest area = 6×6= 24 cm^2(every square is a rectangle)
smallest area= 11×1=11 cm

Answer 2

Consider that a park of length l, breadth b can be made from 24 match sticks.

So we have,

Perimeter = 24 match sticks.

⇒2 ( l + b) = 24

⇒ (l + b) = 12

⇒ l = 12 - b

Now,

Area of the park = l × b

Area of the park (A) = b ( 12 - b) = 12 b - b²

Largest Area :

For maximum area,

A should be maximum, for maxima :

Derivative must be 0.

A' = d/db (12 b - b²)

A' = 12 - 2b

0 = 12 - 2b

12 = 2b

b = 6

So The maximum area of the park possible is 6 ( 12 - 6) = 6² = 36

For minimum value :

Possible values of l, b

l = 12 - b

l, b have to be natural numbers and not equal to 0.

b = 1, l = 12 - 1 = 11, lb = 11

b = 2, l = 12 - 2 = 10 ,lb = 20

b = 3, l = 12 - 3 = 9 , lb = 27

b = 4, l = 12 - 4 = 8 , lb = 32

b = 5, l = 12 - 5 = 7 , lb = 35

b = 6, l = 12 - 6 = 6 ,lb = 36

Therefore, Minimum area of the park will be 11. Maximum area of the park will be 36.