# 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?

From this we get the minimum time as

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,

#T(x)# , that is required to travel from A to C via P.Using

# "speed" = "distance"/"time" # :Along AP the speed is

#4# km/hour, provided#x>0# (otherwise we are going along AB at#5# km/hour) and so:

# " "4 = sqrt(9 + x^2)/t_1 #

# :. t_1 = sqrt(9 + x^2)/4 # Along PC (or AB) the speed is

#5# km/hour, and so:

# " " 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

#x# we get:

# " "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,

#T'(x)=0#

# => 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

#x=4# we have:

# :. 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]}

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To solve the problem, we need to know the distances between the points A, B, and C, and use the formula: Time = Distance / Speed. Without the distances provided, it's not possible to solve the problem. If you have the distances, please provide them, and I can help you calculate the time it takes for Stevie to complete the quest.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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