# Calc

posted by .

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?

• Calc -

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.
.866666.. hrs = 52 minutes , go boom!
.7454 hrs = 44.7 minutes, that would do it
.7333333.. hrs = 44 minutes, so would that.

• Calc -

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.

## Similar Questions

1. ### physics

Two cars are driving toward each other on a straight, flat Kansas road. The Jeep Wrangler is traveling at 84 km/h north and the Ford Taurus is traveling at 43 km/h south, both measured relative to the road. What is the velocity of …
2. ### English

From the thirteenth to the sixteenth century, the Inca empire flourished in the Andes Mountains. It maintained an eleven-thousand-mile network of stone-paved roads. -------------------------------------------------------------------------------- …
3. ### physics

Two cars are driving toward each other on a straight, flat Kansas road. The Jeep Wrangler is traveling at 84 km/h north and the Ford Taurus is traveling at 43 km/h south, both measured relative to the road. What is the velocity of …
4. ### math

in his spare time, kim likes to go cycling. he cycles partly on paved surfaces and partly off-road, through hilly and wooded areas. he cycles at 25km/hon paved surfacesand at 10 km/h off-road. one day, he cycled 41km in 2h. how far …
5. ### Math

A road has an incline of 10°. To the nearest foot, find the increase in altitude of a car after driving 4,000 feet along the road.
6. ### math

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 …
7. ### science physics

A chicken decides to cross a road. The road is perfectly straight, runs north-south, and is full of cars driving by from south to north. The cars have a log-normal distribution for the distance between each car. However, the cars all …
8. ### English

Choose the version of the sentence below that has correctly placed modifiers. A. Coughing and sputtering, Alan steered the old car to the side of the road. B. The old car was coughing and sputtering as Alan steered it to the side of …
9. ### Math

A road rises 1m for every 15m measured along the road. a. What is the angle of inclination of the road to the nearest degree?
10. ### MAths

A four-wheel-drive vehicle is transporting an injured hiker to the hospital from a point that is 30 km from the nearest point on a straight road. The hospital is 70 km down that road from that nearest point. If the vehicle can drive …

More Similar Questions