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?
To determine the shortest possible time our hero has to diffuse the bomb, we need to calculate the time it takes for him to travel through the desert, reach the paved road, and drive to the power plant.
First, let's calculate the time it takes for him to travel through the desert. The distance from his current location to the nearest point on the paved road is 32 km. Since he can travel at a speed of 48 km/hr in the sand, the time it takes to cross the desert is:
Time in desert = Distance / Speed = 32 km / 48 km/hr
Next, let's calculate the time it takes for him to reach the power plant from the point on the paved road. The distance from the paved road to the power plant is 16 km. Since he can travel at a speed of 80 km/hr on a paved road, the time it takes to reach the power plant is:
Time on paved road = Distance / Speed = 16 km / 80 km/hr
Now, let's calculate the total time it takes for him to reach the power plant from his current location:
Total time = Time in desert + Time on paved road
Substituting the values we have:
Total time = (32 km / 48 km/hr) + (16 km / 80 km/hr)
Simplifying the equations:
Total time = (2/3) + (1/5) hours
To add the fractions, we need a common denominator, which is 15:
Total time = (10/15) + (3/15) = 13/15 hours
Now, let's convert the total time to minutes to find out how long our hero has to diffuse the bomb:
Total time in minutes = (13/15) * 60 minutes
= 52 minutes
Therefore, our hero has 52 minutes to diffuse the bomb before it explodes at 12:50 pm.