Week 13 discussion

library(tidyverse)

## Warning: package 'tibble' was built under R version 4.0.4

library(openintro)

Exercise 10

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.

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. \[x^2+y^2=z^2\\ dz/dt=30ft/m \\ x^2 + 10^2 = z^2 \\ and \\ x=sqrt(z^2-100) \]

Take the derivative

\[2xdx/dt=2zdz/dt \] Solve for dx/dt \[dx/dt=z/sqrtz2−100\]

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

b.15 feet out?

z <-15
z/(sqrt(z^2 - 100)) * dz_dt

## [1] 40.24922

c.1 foot from the dock?

z <- 1
z/(sqrt(z^2 - 100)) * dz_dt

## Warning in sqrt(z^2 - 100): NaNs produced

## [1] NaN

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

#answer:We obtain a negative square root