Stevie completes a quest by travelling from #A# to #C# vi #P#. The speed along #AP# is 4 km/hour, and along #AB# it is 5 km/hour. Solve the following?
A) Find #D(x)# a function for the total distance travelled as a function of #x#
B) Form a function #T(x)# for the total journey time.
C) What is the minimum time required for Stevie to complete her quest?
A) Find
B) Form a function
C) What is the minimum time required for Stevie to complete her quest?
1 Answer
From this we get the minimum time as
Explanation:
A) The distance
By Pythagoras;
# \ \ \ \ \ AP^2 = AB^2 + BP^2 #
# :. AP^2 = 3^2 + x^2 #
# :. AP^2 = 9 + x^2 #
# :. \ \ AP = sqrt(9 + x^2) # (must be the +ve root)
Then;
# \ \ \ \ \ D(x) = AP + PC #
# :. D(x) = sqrt(9 + x^2) + (6-x) #
B) Find the time,
Using
Along AP the speed is
# " "4 = sqrt(9 + x^2)/t_1 #
# :. t_1 = sqrt(9 + x^2)/4 #
Along PC (or AB) the speed is
# " " 5 = (6-x)/t_2 #
# :. t_2 = (6-x)/5 #
And so, the total time is given by:
# \ \ \ \ \ T(x) = t_1 + t_2 #
# :. T(x) = sqrt(9 + x^2)/4 + (6-x)/5 # for (#x gt 0# )
C) Stevie's Quest
In order for Stevie to complete the quest we need to find a critical point of T(x):
Differentiating wrt
# " "T'(x) = 1/4*1/2(9+x^2)^(-1/2)*2x+1/5(-1) #
# :. T'(x) = x/(4sqrt((9+x^2))) - 1/5 #
At a critical point,
# => x/(4sqrt(9+x^2)) - 1/5 = 0 #
# :. 5x - 4sqrt(9+x^2) = 0 #
# :. 5x = 4sqrt(9+x^2) #
# :. 25x^2 = 16(9+x^2) #
# :. 25x^2 = 144 + 16x^2 #
# :. 9x^2 = 144 #
# :. x^2 = 144/9 #
# :. x^2 = 16 #
# :. x = 4 # (must be the +ve root)
When
# :. D(4) = sqrt(9 + 16) + (6-4) = 7#
# :. T(4) = sqrt(9 + 16)/4 + (6-4)/5 = 33/20 = 1.65#
We can confirm visually that this corresponds to a minimum by looking at the graph:
graph{sqrt(9 + x^2)/4 + (6-x)/5 [-15, 15, -1, 10]}