calculus

posted by .

2 posts one is 10m high and other is 15m higs stand 30m apart they are to be stayed by transmission wire attached to a single stake at the ground level, the wire running to the top of the post , where should stake place to used least amount of wire ?

  • calculus -

    simplest way:

    draw a parallelogram ABCD, where AB = 10 (first post)
    CD = 15 (2nd post) and BC = 30
    Reflect AB in the line BC to point A1 , so that AB= BA1
    joint DA1 intersecting BC at P

    length of wire = AP+DP = A1P+DP , which is a straight line, and the shortest distance between two points is a straight line, ahhhhh....

    let BP=x
    Triangles ABP and A1PB are congruent, thus AP=A1P.
    Also angles BPA1 and CPD are opposite, thus equal, making triangles APB and DPC similar

    Therefore:
    x/10 = (30-x)/15
    15x = 300-10x
    25x=300
    x = 12

    Put stake 12 m from the 1st post

  • calculus -

    In the above solution, the actual calculation consists of the last 4 lines, the rest explains why it works

    otherwise .... CALCULUS

    AP^2 = x^2 + 10^2
    AP = √(x^2 + 100)

    PD^2 = (30-x)^2 + 15^2
    PD = √(x^2 - 60x + 1125)

    L = √(x^2 + 100) + √(x^2 - 60x + 1125)
    = (x^2 + 100)^(1/2) + (x^2 - 60x + 1125)^(1/2)

    dL/dx = (1/2)(x^2 + 100)^(-1/2) (2x) + (1/2)(x^2 - 60x + 1125)^(-1/2) (5x - 60)
    = 0 for a min of L

    x/√(x^2 + 100) = -(x-30)/√(x^2 - 60x + 1125)
    square both sides

    x^2/(x^2 + 100) = (x^2 - 60x + 900)/(x^2 - 60x + 1125)
    cross multiply
    x^4 - 60x^3 + 900x^2 + 100x^2 - 6000x + 90000 = x^4 - 60x^3 + 1125x^2
    125x^2 + 6000x - 90000=0
    x^2 + 48x - 720 = 0
    (x-12)(x+60) = 0
    x = 12 or x = -60 , but x can't be negative, so

    x = 12

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. geometry

    A guy wire is used to stabilize a utility pole. The wire is fastened to the pole 25 feet above the ground, and staked into the ground 4 feet from the base of the pole. (No stand-offs are used, the wire goes directly from the pole to …
  2. Calculus

    Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to single stake, running from ground level to the top of each post. Where should the stake be placed to use the …
  3. Calculus

    Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to single stake, running from ground level to the top of each post. Where should the stake be placed to use the …
  4. Math

    A wire is stretched from the top of a 12-foot pole to a stake in the ground and then to the base of the pole. If a total of 20 feet of wire is needed, how far is the stake from the pole (the answer is NOT 16)
  5. Trig

    A guy wire attached to the top of an electric pole makes a 700 angle with the level ground. At a point 25 feet from the guy wire (farther away from the pole), the angel of elevation to the top of the pole is 420. How long is the guy …
  6. Trig

    A guy wire attached to the top of an electric pole makes a 70deg angle with the level ground. At a point 25 feet from the guy wire (farther away from the pole), the angel of elevation to the top of the pole is 42deg. How long is the …
  7. math

    Find the length of a guy wire that makes an angle of 45.5 degrees with the ground if the wire is attached at the top of a tower 63.0 meters high
  8. math

    A rectangular savannah is 30m long by 15m wide. The Ministry of Education wants to fence that savannah using posts and wire. If the posts, costing $18.00 each, are placed 5m apart andt the wire cost $35.00 per metre. Calculate the …
  9. math

    A rectangular savannah is 30m long by 15m wide. The Ministry of Education wants to fence that savannah using posts and wire. If the posts, costing $18.00 each, are placed 5m apart andt the wire cost $35.00 per metre. Calculate the …
  10. maths

    two poles 6m and 15m tall are 20 m apart a wire is attached to the top of each pole and is also staked to the ground some where b/n two poles where should be the wire be staked so that the minimum amount of wire be used?

More Similar Questions