Nitin wants to construct a rectangular plastic tank for his house that can hold 80 ft 3 of water. The top of the tank is open. The width of tank will be 5 ft but the length and heights are variables. Building the tank cost Rs.20 per sq. foot for the base and Rs.10 per sq. foot for the side
Answers
Given : The top of the tank is open. The width of tank will be 5 ft but the
length and heights are variables.
Building the tank cost 20 per sq. foot for the base and 10 per sq. foot for the side.
Volume of tank is 80 cubic ft
To Find : In order to make a least expensive water tank, Nitin need to minimize what ?
Solution:
In order to make a least expensive water tank, Nitin need to minimize cost
The width of tank will be 5 ft
Let sat height = h
Volume = length * width * height
=> 80 = 5 * length * h
=> length = 16/h
Base area = 5 * 16/h
CSA = 2 ( 5 + 16/h) * h
Cost = 20 ( 5 * 16/h ) + 10 * 2 ( 5 + 16/h) * h
= 1600/h + 100h + 320
c(h) = 100h + 320 + 1600/h
Range of h (0 , ∞)
c(h) = 100h + 320 + 1600/h
=> c'(h) = 100 - 1600/h²
c'(h) = 0
=> 1600/h² = 100
=> h = 4
c''(h) = 3200/h³ > 0
Hence cost is minimum at h = 4
The cost of least expensive tank
c(h) = 100h + 320 + 1600/h
c(4) = 400 + 320 + 400
= 1120
Learn More:
A sheet of area 40 m2 in used to make an open tank with a square ...
brainly.in/question/10126107
A square of x cm is cut from each corner of a square metal sheet of ...
brainly.in/question/13516236
A open box with a square base is to be made. Its total surface area is ...
brainly.in/question/11774623