Question 314.
A rectangular garden, area of 300 m^2. The cost of fencing three sides is $9 /m and the cost of fencing the fourth side is $15/m. Find the dimensions of the garden such that the cost of fencing is minimum and the minimum cost of fencing the garden.
Answer 314.
Let the two adjacent sides of the fence be of lengths = a and b in m.
=> ab = 300 ... ( 1 ) and
Cost of fencing,
C = 9 * (2a + b) + 15b = 18a + 24b
Plugging a = 300/b in the equation of cost,
C = 5400/b + 24b
For C to be minimum, dC/db = 0 and d^2C/db^2 > 0
dC/db = 0 => - 5400/b^2 + 24 = 0 => b = 15 m and a = 20 m
d^2C/db^2 = 10800/b^3 > 0
=> Minimum cost of fencing will be when one side is 15 m and the other 20 m with cost of fencing one 15 m side being $ 15/m and for the remaining three sides $ 9/m
and the minimum cost
= $ (18* 20 + 24 * 15)
= $ 720.
Link to YA!
A rectangular garden, area of 300 m^2. The cost of fencing three sides is $9 /m and the cost of fencing the fourth side is $15/m. Find the dimensions of the garden such that the cost of fencing is minimum and the minimum cost of fencing the garden.
Answer 314.
Let the two adjacent sides of the fence be of lengths = a and b in m.
=> ab = 300 ... ( 1 ) and
Cost of fencing,
C = 9 * (2a + b) + 15b = 18a + 24b
Plugging a = 300/b in the equation of cost,
C = 5400/b + 24b
For C to be minimum, dC/db = 0 and d^2C/db^2 > 0
dC/db = 0 => - 5400/b^2 + 24 = 0 => b = 15 m and a = 20 m
d^2C/db^2 = 10800/b^3 > 0
=> Minimum cost of fencing will be when one side is 15 m and the other 20 m with cost of fencing one 15 m side being $ 15/m and for the remaining three sides $ 9/m
and the minimum cost
= $ (18* 20 + 24 * 15)
= $ 720.
Link to YA!
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