Page 179 Question 10

A boat is being pulled into a dock at a constant rate of 30ft/min by a winch located 10ft above the deck of the boat.

At what rate is the boat approaching the dock when the boat is:

  1. 50 feet out?

  2. 15 feet out?

  3. 1 foot from the dock?

  4. What happens when the length of rope pulling in the boat is less than 10 feet long?

Part A

The picture in the book shows us that the Boat, dock and rope form a right triangle. Where the rope is the hypotenuse.

Because of this we can use Pythagorean theorem which states \(a^2+b^2=c^2\)

a = distance of the boat from the dock

b = height of winch above the water (10 ft)

c = length of the rope

The problem tells us that the boat is being pulled at a constant rate of 30ft/min so \(\frac{dy}{dt} = 30 ft/min\)

We want to find \(\frac{da}{dt}\) when a = 50

Differentiate with respect to t

\(2a \frac{da}{dt} + 2b \frac{db}{dt} = 2c \frac{dc}{dt}\)

b is a constant so it = 0

\(2a \frac{da}{dt} = 2c \frac{dc}{dt}\)

\(a \frac{da}{dt} = c \frac{dc}{dt}\)

Now, solve for \(\frac{da}{dt}\)

\(\frac{da}{dt} = \frac{c * \frac{dc}{dt}}{a}\)

We can now write R code to solve this for us:

# Constants
b <- 10
dc_dt <- 30  # The rope is being pulled in at 30 ft/min

# Function to calculate the rate of approach of the boat
calculate_da_dt <- function(a) {
  c <- sqrt(a^2 + b^2)
  da_dt <- (c * dc_dt) / a
  return(da_dt)
}
# 50 feet out
part_a <- calculate_da_dt(50)
print(part_a)
## [1] 30.59412

When the boat is 50 feet out it is approaching the dock at 30.59412 ft/min

Part B

# 15 feet out
part_b <- calculate_da_dt(15)
print(part_b)
## [1] 36.05551

When the boat is 15 feet out it is approaching the dock at 36.05551 ft/min

Part C

# 15 feet out
part_c <- calculate_da_dt(1)
print(part_c)
## [1] 301.4963

When the boat is 1 foot out it is approaching the dock at 301.4963 ft/min

Part D

Since the winch is located 10 feet above the deck of the boat if the rope were to be less than 10 feet long then the rope would not be long enough to reach the boat while it was in the water so the boat would either have to have part of the front lifted above the water or it would simply just not be possible without other mechanisms in place.