A boat is being towed by a rope that goes from water level of the boat to a pulley on the dock...?

first, great produce sketch of dock, boat, rope , water level.
assume rope attached boat @ water-line of boat.

height of dock (where pulley is) 5 feet higher boat is.
let boat s feet dock / pulley.
so, wall of dock, water-level , rope form right-angled triangle.
let angle between rope , surface of water = α.
have: tanα = 5/s, get: cosα = s / √(5² + s²) {from pythagoras}.
i.e.: cosα = s / √(25 + s²).

let speed of rope, in direction rope moving = v.
horizontal component of velocity of rope = v.cosα.
therefore, horizontal component of velocity of rope = v.s / √(25 + s²),
, this, definition, must speed boat approaches dock / pulley.

so, putting in 'the numbers':
speed of boat in water when boat 12 feet dock / pulley
= 2 x 12 /√(25 + 144) = 24 / 13 = 1.85 feet / sec.

hth,
skywave.

the pulley 5 ft higher water level. rope being pulled in @ rate of 2 ft/sec. find rate @ boat approaching dock when boat 12 ft dock.

Science & Mathematics Mathematics Next