A load must be suspended 6 meters below a high ceiling using cables attached to
ID: 2864385 • Letter: A
Question
A load must be suspended 6 meters below a high ceiling using cables attached to two supports that are 2 meters apart (see figure). Assume that the point where the cables are joined is equally distant from each support. How far below the ceiling (labeled x) should the cables be joined to minimize the total length of cable used? What is the minimum amount of cable needed? (Write your answer in EXACT form.)
(a) To minimize, x = ?
A load must be suspended 6 meters below a high ceiling using cables attached to two supports that are 2 meters apart (see figure). Assume that the point where the cables are joined is equally distant from each support. How far below the ceiling (labeled x) should the cables be joined to minimize the total length of cable used? What is the minimum amount of cable needed? (Write your answer in EXACT form.) (a) To minimize, x = ? (b) Minimum total amount of cable = ?Explanation / Answer
let L be the length of wire used.
Let x be the distance between the ceiling and the point where the cables are joined.
If you draw a picture, you will see that there are 2 lengths of wire sqrt(1 + x^2) m long and one that it (6 - x) m long.
So, the length L of wire used is:
L = 2sqrt(1 + x^2) + 6 - x
differentiate both sides wrt x
L' = 2x/sqrt(1 + x^2) - 1
set L' = 0
2x/sqrt(1+x^2) = 1
square both sides
4x^2/(1+x^2) = 1
3x^2 = 1
x = 1/sqrt(3) = 0.58m
therefore, the wires should be joined 0.58m below the ceiling.
x = 0.58 m.....answer for (a)
minimum total amount of cable is at x = 0.58 in L
L = 2sqrt(1 + 0.58^2) + 6 - 0.58
L = 7.732 m.......answer for (b)
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