A power company needs to lay a cable from point A on one bank of an 800-foot wide,
straight river to point B on the opposite bank 1600 feet downstream. The power
company will choose a point C on the same side of the river as B, and will lay a cable
underwater from A to C and on land from C to B. It costs $3 a foot to lay the cable on
land and $5 a foot to lay it underwater. Where should point C be chosen to minimize the
total cost and what is the minimum cost?
PLEASE SHOW FULL DERIVATIVE STEPS! THANK YOU
straight river to point B on the opposite bank 1600 feet downstream. The power
company will choose a point C on the same side of the river as B, and will lay a cable
underwater from A to C and on land from C to B. It costs $3 a foot to lay the cable on
land and $5 a foot to lay it underwater. Where should point C be chosen to minimize the
total cost and what is the minimum cost?
PLEASE SHOW FULL DERIVATIVE STEPS! THANK YOU
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A
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D ...C ...B
we shall solve in general form & then substitute numerical values
let AD = w
DC = x
DB = y
cost ratio/ft water:land = k (5/3)
angle DAC = z
dist. over water = w•sec z,
dist. over land = y - w•tan z
cost C ∞ k•sec z +(y - tan z)
dC/dz = k•secz•tan z - sec^2 z
= (ksin z - 1) / cos^2 z
for minima, sin z = 1/k
which yields x = w /√(k^2 - 1) = 800/sqrt((5/3)^2-1) = 600 ft <-------
cost can now be easily worked out as 1000*5+1000*3=$8000 <-------
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D ...C ...B
we shall solve in general form & then substitute numerical values
let AD = w
DC = x
DB = y
cost ratio/ft water:land = k (5/3)
angle DAC = z
dist. over water = w•sec z,
dist. over land = y - w•tan z
cost C ∞ k•sec z +(y - tan z)
dC/dz = k•secz•tan z - sec^2 z
= (ksin z - 1) / cos^2 z
for minima, sin z = 1/k
which yields x = w /√(k^2 - 1) = 800/sqrt((5/3)^2-1) = 600 ft <-------
cost can now be easily worked out as 1000*5+1000*3=$8000 <-------