Show work where needed. Find the dimensions of the rectangular region of area 20

Question

Show work where needed.

Find the dimensions of the rectangular region of area 200 in2 with least perimeter. It costs $8 per linear foot of fencing" What is the maximum rectangular area that can be fenced for $4000? Assume the cost of constructing an oil transportation pipeline is $500,000 per kilometer on land and $1,800,000 per kilometer under water. Find the route that minimizes the cost of construction of a pipeline from a rig that is 10 km off-shore to a refinery that is 5 km down shore. Find the dimensions and the maximum volume of a open rectangular box that can be constructed from a 18 x 42 rectangular piece of cardboard by removing four squares at the four corners.

Explanation / Answer

1.

let a, b be sides of the rectangle , then ab =200,

then perimeter = 2(a+b) ,

we know that (a+b)>=2(ab)^0.5, equality when a=b,

for maximum value of perimeter, a=b

=> a = b = 10*(2)^0.5 => max perimter = 40*(2)^(0.5) = 56.56

2.

perimter that can be fenced = 4000/8 = 500

=> 2(a+b) = 500

=> a+b = 250

we know that (a+b)>=2(ab)^0.5, equality when a=b,

=> maximum area when a=b,

=> a= b= 125 => max area = 15625

4.

let the side be a,

then volume V= a(42-2a)(18-2a) = 4a(a-21)(a-9)

V' = 4[(a-21)(a-9)+a(a-9)+a(a-21)] = 4[3a^2-60a+189] =12[a^2-20a+63 ]=0 when a = 10-37^0.5 = 3.92

dimensions = 34.16x10.16x3.92

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