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steposvetlana [31]
3 years ago
13

given line segment jl with j(8 -8) and l(-16 -2) find the coordinates of K if K divides line segment JL two-thirds of the way fr

om J to L

Mathematics
1 answer:
Masja [62]3 years ago
4 0

Answer:

  • (-8, -4)

Step-by-step explanation:

<h3>Given</h3>
  • Line segment JL with J(8, -8) and L(-16, -2)
<h3>To find</h3>
  • Coordinates of K if K divides JL two-thirds from J
<h3>Solution</h3>
  • JK = 2/3JL

<u>Coordinates of K</u>

  • x = 8 + 2/3(-16 - 8) = 8 + 2/3(-24) = 8 - 16 = - 8
  • y = -8 + 2/3(-2 - (-8)) = -8 + 2/3(6) = -8 + 4 = - 4
  • K = (-8, -4)

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A cable television compan is laying cable in an area with underground utilities. Two subdivisions are located on opposite sides
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Answer:

Step-by-step explanation:

Given:

Point Q = Q is on the north bank 1200 m east of P $40/m to lay cable underground and $80/m way to lay the cable.

Lets denote T be a point on north bank, therefore, point T is t m east and 100m north of P.

Note that 0 < t < 1200

Distance from P to T using Pythagoras theorem:

PT^2 = PQ^2 + QT^2

PT = √(t²+100²) = √(t²+10,000)

Distance from T to Q:

TQ = 1200 - t

From above, Cable from P to T is underwater, and costs $80/m to lay.

Cable from T to Q is underground, and costs $40/m to lay.

Total cost of the cable:

C(t) = 80√(t²+10,000) + 40(1200 - t)

Cost is at its minimum: when dC/dt = C'(t) = 0 and d^2C/dt^2 = C''(t) > 0

dC/dt = C'(t)

= 80 * 1/2 (t² + 10,000)^(-1/2) × 2t + 40 × (-1)

dC/dt = C'(t)

= 80t/√(t² + 10,000) - 40 = 0

80t/√(t² + 10,000) = 40

80t = 40√(t² + 10,000)

2t = √(t² + 10,000)

square both sides:

4t² = t² + 10,000

3t² = 10,000

t² = 10,000/3

t = 100/√3

≈ 57.735

d^2C/dt^2 =

C''(t) = [80√(t² + 10,000) - 80t × t/√(t²+10,000)] /√(t²+10,000)

C''(t) = [(80(t²+10,000) - 80t²) / √(t²+10,000)] /√(t²+10,000)

C''(t) = 800,000 / (t² + 10,000)^(3/2)

C''(t) > 0 for all t

Therefore C(t) is minimized at t = 100/√3

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= 80 √(40,000/3) + 48,000 - 4000/√3

= 16000/√3 + 48,000 - 4000/√3

= 48,000 + 12,000/√3

= 48,000 + 4,000√3

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Minimum cost is $54,928.20

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