18.The government of a state which has mostly hilly area decided to have adventurous playground on the top of hill having plane area and space for 10000 persons to sit at a time.After survey it was decided to have rectangular playground with a semicircular parking at one end of playground only as space is less.The total perimeter of the field is measured as 1000m as shown.
Answers
Given : rectangular play ground with a semicircular parking at one end of play ground
The total perimeter of the field is measured as 1000 m
One side is 2y other side is x , semicircle on side 2y
To Find : the relation between x and y
Area of sports ground in terms of x
Solution:
side = 2y = diameter
=> radius = y
Hence semicircle perimeter = πy
Total perimeter = x + 2y + x + πy
= 2x + 2y + πy
Total perimeter = 1000 m
=> 2x + 2y + πy = 1000
Area of sports ground= x(2y)
2x + 2y + πy = 1000 => y = (1000 - 2x) /( π + 2)
Area = x * 2 (1000 - 2x) /( π + 2)
= (2/( π + 2) ( 1000x - 2x²)
The maximum area of sports ground is for x equal to
dA/dx = (2/( π + 2) ( 1000 - 4x)
=> dA/dx = 0 if x = 250
d²A/dx² = (2/( π + 2) (-4) = -ve < 0
Hence maximum area of sports ground is for x equal to 250
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