# Thorsten the geologist is in the desert, 10 km from a long, straight road. On the road, Thorsten's jeep can do 50kph, but in the desert sands, it can manage only 30kph. How many minutes will it take Thorsten to drive through the desert? (See details).

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Thorsten is very thirsty, and knows that there is a gas station 25 km down the road (from the nearest point N on the road) that has ice-cold Cola.

A diagram of the problem:

(a) How many minutes will it take for Thorsten to drive to P through the desert?

Rate = Distance/time

Time = Distance/rate

Time = PA/30 kph

PA can be solved with Pythagorean theorem.

(b) Would it be faster if Thorsten first drove to N and then used the road P?

Found the time to get to N from A, then the time from N to P using the same techniques above.

(c) Find an even faster route for Thorsten to follow. Is your route the fastest possible?

???

Thorsten is very thirsty, and knows that there is a gas station 25 km down the road (from the nearest point N on the road) that has ice-cold Cola.

A diagram of the problem:

(a) How many minutes will it take for Thorsten to drive to P through the desert?

Rate = Distance/time

Time = Distance/rate

Time = PA/30 kph

PA can be solved with Pythagorean theorem.

(b) Would it be faster if Thorsten first drove to N and then used the road P?

Found the time to get to N from A, then the time from N to P using the same techniques above.

(c) Find an even faster route for Thorsten to follow. Is your route the fastest possible?

???

##### 1 Answer

(a)

#### Explanation:

(a) As

and it will take

or

(b) If Thorsten first drove to N and then used the road P,

he will take

and he will be faster.

(c) Let us assume he directly reaches

then

To find extrema, let us differentiate w.r.t.

or

or

i.e.

and time taken will be

=