It is noon. A secret agent is driving a jeep through the desert in the tiny principality of Alta Loma. He is 32km from the nearest point on a straight, paved road. Down the road 16km is a power plant in which a band of international terrorists has placed a time bomb set to explode at 12:50pm. The jeep can travel at 48km/hr in the sand at 80 km/hr on a paved road. If he arrives at the power plant in the shortest possible time, how long will our hero have to diffuse the bomb?

Geesh, I wish authors would stick to the traditional wording of the "cable through a river" problem, instead of trying to be "current"

anyway...

Make a sketch labeling his position B and the point on the road 32 km away as A. let the point 16 km down from A be C
Suppose he aims for a point on the road between A and C, call that P, let AP=x
So his path is BP +PC
then BP^2 = x^2 + 32^2
BP = (x^2 + 1024)^(1/2)
time to go BP = (x^2 + 1024)^(1/2)/48
time to go PC = (16-x)/80

T = (x^2+1024)^(1/2) /48+ 16/80 - x/80
dT/dx = (1/96)(x^2 + 1024)^(-1/2) (2x) - 1/80
= 0 for a min of T
2x/(96√(x^2+1024)) = 1/80
96√(x^2+1024) = 160x
3√(x^2+1024) = 5x
square both sides
9(x^2 + 1024) = 25x^2
9216 = 16x^2
x^2 = 576
x = √576 = 24

But we clearly expected x to be between 0 and 16, so let's investigate the two direct routes.
1. directly to A, then AP
T = 32/48 + 16/80 = .86666 hrs
2. directly to C
distance = √(16^2 + 32^2) = √1280
T = √1280/48 = .7454

3. If we "blindly" sub in our x = 24 we get a total time
of T = √1600/48 + (-8/80) = 11/15 = .73333

3. consider a point between A and C, say x = 4
distance through sand = √(32^2+4^2) = √1040
time through sand = √1040/48 = .67185 hrs
time along road = 12/80 = .15
total time = .82185

I am puzzles by these results, and I can only guess that I made some arithmetic error somewhere. Just can't seem to find it, perhaps somebody else can find it.

anyway , he has 50 minutes to get there.
my 3 answers are :
.866666.. hrs = 52 minutes , go boom!
.7454 hrs = 44.7 minutes, that would do it
.7333333.. hrs = 44 minutes, so would that.

let me take a whack at it.

If the jeep arrives at the road a distance x from the bomb, then

the distance on road is x
distance on sand is √(32^2 + (16-x)^2)

travel time is thus

t = x/80 + √(32^2 + (16-x)^2)/48
dt/dx = 1/80 - (16-x)/48√(x^2-32x+1280)

dt/dx = 0 when x = -8
Also puzzling. And the graph of t(x) appears not to have a minimum for x>0.

To determine the shortest possible time for the secret agent to reach the power plant, we need to calculate the time it takes to travel through both the desert and the paved road.

Step 1: Calculate the time to travel through the desert:
Since the secret agent is 32km away from the nearest point on the straight, paved road, he needs to travel this distance at a speed of 48km/hr. We can use the formula: time = distance / speed.
Time in the desert = 32km / 48km/hr = 2/3 hour or 40 minutes.

Step 2: Calculate the time to travel on the paved road:
Since the power plant is 16km away from the starting point on the paved road, the secret agent needs to travel this distance at a speed of 80km/hr.
Time on the paved road = 16km / 80km/hr = 1/5 hour or 12 minutes.

Step 3: Calculate the total time to reach the power plant:
Total travel time = Time in the desert + Time on the paved road
Total travel time = 40 minutes + 12 minutes
Total travel time = 52 minutes.

Finally, since the secret agent is starting at noon and it will take 52 minutes to reach the power plant, he will have 12:50pm - 52 minutes = 11:58am to diffuse the bomb.

To determine the shortest possible time for the secret agent to reach the power plant and diffuse the bomb, we need to calculate the time it takes for him to travel on both the sand and paved road.

First, let's calculate the time it takes for him to travel on the sand:

Distance from the jeep to the nearest point on the paved road: 32 km
Speed on sand: 48 km/hr

Time taken on sand = Distance / Speed = 32 km / 48 km/hr = 0.67 hr

Next, let's calculate the time it takes for him to travel on the paved road:

Distance from the nearest point on the paved road to the power plant: 16 km
Speed on a paved road: 80 km/hr

Time taken on the paved road = Distance / Speed = 16 km / 80 km/hr = 0.2 hr

Finally, let's calculate the total time taken to reach the power plant:

Total time taken = Time taken on sand + Time taken on paved road = 0.67 hr + 0.2 hr = 0.87 hr

Since the secret agent starts at noon, the time available to diffuse the bomb is:

Time available = 12:50 pm - 12:00 pm = 0.83 hr

Therefore, our hero will have approximately 0.83 hours (or 50 minutes) to diffuse the bomb once he arrives at the power plant.