A new machine that deposits cement for a road requires 13 hours to complete a one-half mile section of road. An older machine requires 15 hours to pave the same amount of road. After depositing cement for 2 hours, the new machine develops a mechanical problem and quits working. The older machine is brought into place and continues the job. How long does it take the older machine to complete the job
13h/0.5mi. = 26 h/mi. with new machine.
d1 = 2h * 1mi/26h = 2/26= 1/13 Mi. by new machine.
d2 = 13/26 - 2/26 = 11/26 mi. remaining.
15h/0.5mi. = 30 h/mi. with old
machine.
T = (11/26)mi * 30h/mi = 12.69 h.
To determine how long it takes the older machine to complete the job, we can use the concept of work rates.
Let's start by calculating the work rates of the two machines.
The new machine takes 13 hours to complete a one-half mile section of road. Therefore, its work rate is:
1 ÷ 13 = 1/13 mile per hour
Similarly, the older machine takes 15 hours to complete a one-half mile section of road. Therefore, its work rate is:
1 ÷ 15 = 1/15 mile per hour
Now, let's consider the scenario where the new machine runs for 2 hours before developing a mechanical problem.
In those 2 hours, the new machine has completed:
2 hours × (1/13 mile per hour) = 2/13 mile
Since the new machine quits working at this point, the remaining distance to be paved is:
1/2 mile - 2/13 mile = 6/13 mile
Now, to find out how long it takes the older machine to complete the remaining 6/13 mile, we'll use the work rate of the older machine.
The time required by the older machine can be calculated as:
(6/13 mile) ÷ (1/15 mile per hour) = (6/13) × (15/1) = 90/13 hours ≈ 6.92 hours
Therefore, it takes approximately 6.92 hours (or about 6 hours and 55 minutes) for the older machine to complete the job after taking over from the new machine.