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 play ground with a semicircular
parking at one end of play ground only as space is less. The total perimeter of the field is
measured as 1000 m as shown
Based on the above information answer the following.
(i) Looking at the figure (plan) the relation between x and y is
a. x + 2y + πy = 1000
b. 2x + 2y + πy = 1000
c. x + 2y + πy = 500
d. x + y + πy = 1000
(ii) Area of sports ground in terms of x is
a.
2
π+2
(1000x − 2x
2
)m2
b. 1
π
(1000x − 2x
2
)m2
c.
2
π+2
(500x − 2x
2
)m2
d. 1
π
(500x − 2x
2
)m2
(iii) The maximum area of sports ground is for x equal to
a. 500m
b. 50m
c. 100m
d. 250m
(iv) The government wants to maximize the area including parking area for this to
happen, value of y is
a.
1000
4+π
m
b. 2000
4+π
m
c.
500
4+π
m
d. 750
4+π
m
Answers
Answer:
Please find tha attachment
:
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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