Week 13 Discussion: Applications of the Derivative and Integration

Using R, provide the solution for any exercise in either Chapter 4 or Chapter 7 of the calculus textbook. If you are unsure of your solution, post your concerns.

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

\[x^2 + y^2 = z^2, \frac{dz}{dt}=\frac{30ft}{m}\]

\[x^2 + 10^2 = z^2\] and \[x=\sqrt{z^2-100}\]

Take the derivative \[2x\frac{dx}{dt}=2z\frac{dz}{dt}\] Solve for dx/dt \[\frac{dx}{dt}=\frac{z}{\sqrt{z^2-100}}\frac{dz}{dt}\]

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

  1. 50 feet out?
z <-50
dz_dt <- 30
z/(sqrt(z^2 - 100)) * dz_dt
## [1] 30.61862
  1. 15 feet out?
z <-15
z/(sqrt(z^2 - 100)) * dz_dt
## [1] 40.24922
  1. 1 foot from the dock?
z <- 1
z/(sqrt(z^2 - 100)) * dz_dt
## Warning in sqrt(z^2 - 100): NaNs produced
## [1] NaN
  1. What happens when the length of rope pulling in the boat is less than 10 feet long? We obtain a negative square root