Math(module 4 Calculus)

posted by .

(minimum commuting time) A lighthouse lies 2 miles offshore directly across from point A of a straight coastline. The lighthouse keeper lives 5 miles down the coast from point A.What is the minimum time it will take the lighthouse keeper to commute to work, rowing his boat at 3 miles per hour, and walking at 5 miles per hour?

  • Math(module 4 Calculus) -

    Let the distance from A to his landing place be x
    then from there it would be another 5-x miles to his house.
    let the distance he rows be d
    d^2 = x^2 + 2^2
    d = (x^2 + 4)^(1/2)

    time for the rowing part = (x^2 + 4)^(1/2) /3
    time for the walking part = (5-x)/5

    Time = (x^2 + 4)^(1/2) /3 + (5-x)/5
    d(Time)/dx = (1/6)(x^2 + 4)^(-1/2) (2x) - 1/5
    x/(3√(x^2+4) ) - 1/5
    = 0 for a minimum of Time.

    x/(3√(x^2+4) ) - 1/5
    5x = 3√(x^2 + 4)
    25x^2 = 9(x^2+4)
    25x^2 = 9x^2 + 36
    16x^2 = 36
    4x = 6
    x = 6/4

    sub that into Time = .. and you got it
    ( I got 1 hour and 32 minutes)
    check my arithmetic

  • Math(module 4 Calculus) -

    yoo

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calculus

    A lighthouse is built on an exposed reef, 5.5 miles off-shore. The shoreline is perfectly straight, and a town is located 6.5 miles downshore from the point on the shoreline nearest the lighthouse. The lighthouse keeper needs to go …
  2. calculus

    A lighthouse is built on an exposed reef, 5.5 miles off-shore. The shoreline is perfectly straight, and a town is located 6.5 miles downshore from the point on the shoreline nearest the lighthouse. The lighthouse keeper needs to go …
  3. pre calc

    Lighthouse B is 8 miles west of lighthouse A. A boat leaves A and sails 5 miles. At this time, it is sighted from B. If the bearing of the boat from B is N63E, how far from B is the boat?
  4. Calculus

    A man is in a boat 4 miles off a straight coast. He wants to reach a point 10 miles down the coast in the least possible time. If he can row 4 miles per hour and run 5 miles per hour, where should he land the boat?
  5. Math

    A LIGHTHOUSE IS 10 MILES NORTHEAST OF A DOCK. A SHIP LEAVES THE DOCK AT NOON,AND SAILS EAST AT A SPEED OF 12 MILES AN HOUR. AT WHAT TIME WILL IT BE 8 MILES FROM THE LIGHTHOUSE?
  6. pre-calc

    two lighthouses A and B are positioned along the coast.A ship is positioned 30 miles from lighthouse A and 40 miles from lighthouse B.The angle between the line of sight from the ship to lighthouse A and the line of sight to lighthouse …
  7. Calculus

    You are in a boat "a" miles from the nearest point on the coast. You are to go to a point Q, which is "b" miles down the coast and 1 mile inland. You can row at 2 miles per hour and walk at 4 miles per hour. If a=3, and b=4, toward …
  8. Calculus

    A lighthouse is built on an exposed reef, 5 miles off-shore. The shoreline is perfectly straight, and a town is located 9 miles downshore from the point on the shoreline nearest the lighthouse. The lighthouse keeper needs to go from …
  9. Calculus (trig derivatives)

    A lighthouse 2 miles offshore has a light that rotates once every 20 seconds. At what rate is the light traveling along the shore if it is shining on a point 4 miles along the shore from the point nearest the lighthouse?
  10. Calculus

    A light in a lighthouse 5 kilometers offshore from a straight shoreline is rotating at 4 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point 5 kilometers from the point opposite the‚Äč lighthouse?

More Similar Questions