SOLUTION: A scientist wants to go to a small island 5km south of the coastline and 13km from the town. From town, he drives his small truck east along the waterfront, towing his boat. Then

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Question 453838: A scientist wants to go to a small island 5km south of the coastline and
13km from the town. From town, he drives his small truck east along the
waterfront, towing his boat. Then he parks his truck, and travels straight to
the island by boat.

(a) The truck travels at 50km/h. The boat travels at 20km/h. How far east
of town should he park his truck and start his boat journey to minimize
the travel time?
(b) The truck uses $1 of diesel per km travelled, while the boat uses 60c of
diesel per km. What should the scientist do if he wants to minimize the
cost?

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
A scientist wants to go to a small island 5km south of the coastline and
13km from the town.
From town, he drives his small truck east along the waterfront, towing his boat.
Then he parks his truck, and travels straight to the island by boat.
:
(a) The truck travels at 50km/h. The boat travels at 20km/h.
How far east of town should he park his truck and start his boat journey to minimize the travel time?
:
Find the distance (d) to the point on the shore just north of the island
d = sqrt%2813%5E2-5%5E2%29
d = 12 km to a point on the shore due north of the island, (from the town)
:
Let p = distance to point on the shore east of town to launch the boat
then
(12-p) = distance from that point to a point due north of the island
:
The boat route to the island is the hypotenuse of a right angle where:
a = 5
b = (12-p)
:
Write a time equation, time = dist/speed
total time (t) for the truck and boat
:
total time = truck time + boat time
t = p%2F50 + %28sqrt%285%5E2%2B%2812-p%29%5E2%29%29%2F20
:
t = p%2F50 + %28sqrt%2825%2B%28144-24p+%2B+p%5E2%29%29%29%2F20
:
t = p%2F50 + %28sqrt%2825%2B144-24p+%2B+p%5E2%29%29%2F20
:
t = p%2F50 + %28sqrt%28p%5E2-24p%2B169%29%29%2F20
Graphing this

total travel time minimum occurs when:
p = 9.8 km by truck to a point east of town, launch the boat there
:
:
:
(b) The truck uses $1 of diesel per km traveled, while the boat uses 60c of
diesel per km.
What should the scientist do if he wants to minimize the cost.
Modify the equation to:
Cost = 1p + .6%2Asqrt%28p%5E2-24p%2B169%29
Graph this
+graph%28+300%2C+200%2C+-6%2C+5%2C+-10%2C+10%2C+x%2B.6sqrt%28x%5E2-24x%2B169%29%29+
It appears he should not drive the truck at all but travel all the way by boat (p=0).
His expense the will be 13*.6 = $7.80, will increase for each km he drives the truck