Question

A small island is 4 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 4 miles per hour and can walk 5 miles per hour, where should the boat be landed in order to arrive at a town 7 miles down the shore from P in the least time? Let x be the distance (in miles) between point P and where the boat lands on the lakeshore.
(a) Enter a function T(x) that describes the total amount of time the trip takes as a function of the distance x .
T(x) =________
(b) What is the distance x = c that minimizes the travel time?
c =_______
(c) What is the least travel time?
The least travel time is_________.

Answer 1

Answer:

a) t(x)  = [ √ (4)² + (x)²]/ 4    +   7/5  - x/5

b) x = 0,97 miles

c) t (min)  = 1,24 hours

Step-by-step explanation: See Annex

Figure in annex shows a clear description of the situation

Let  x be distance in miles between point P and where boat lands

A woman has to row  a distance L

L = √ (4)² + (x)²

and the part to get to the town (which she has to walk)

d = ( 7 -  x)

But we are required to give time as a function of x

distance   =  speed * time     ⇒   t  = distance / speed

Therefore

t(x)  = [ √ (4)² + (x)²]/ 4   +   ( 7  -  x  ) / 5

t(x)  = [ √ (4)² + (x)²]/ 4    +   7/5  - x/5

Taking derivatives both sides of the equation

t¨(x)   = ( 2x)*4 / 16√ (4)² + (x)²]   -  5/25  

t¨(x)   = x/  2√ (4)² + (x)²]  - 1/5

t¨(x)   = 0      x/  2√ (4)² + (x)²]  - 1/5   =  0

[ 5 x  -  2√ (4)² + (x)²]  =  0       ⇒   5x    =  2√ (4)² + (x)²]   or

   25 x² =  4  (  4 +   x²)

   25 x² = 16  +   8x²

  17x²   =   16

  x²  =   16/17      

   x²   =  0,941

  x = 0,97 miles  

And the distance walking to get to town

d =  7  -  x       d =   7  -  0,97

d  = 6,03 miles

The least travel time  is   t(x)  = [ √ (4)² + (x)²]/ 4    +   7/5  - x/5  

t (min)  =  √ 16 +  0,94) /4   +  1,4  - 0,97/5

t (min)  = 1,03  +  1,4  - 0,194

t (min)  = 1,24 hours