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? (Round your answer to one decimal place.)

1 h

d=r1*t+r2*t

job=job/13 hrs*timetotal1+job/15*timetotal2

but time total2=2 hrs
time total1 is what is to be determined

1=1/15* 2 + 1/13 * (t-2) where t is the time after the two hours left.

13*15=26+ 15t-30
15t=13*15-4
t= 12.7333 hrs check my thinking

The new machine completes 2/13 of the job.

So, the older machine requires (11/13)/(1/15) = 12.7 hours to do the rest.

To find the time it takes for the older machine to complete the job, we need to determine the amount of work done by the new machine in 2 hours and subtract it from the total work required.

The new machine takes 13 hours to complete a one-half mile section of road, so it does 1/13 of the work in 1 hour.

In 2 hours, the new machine does 2/13 of the work.

Therefore, the remaining work to be done is 1/2 - 2/13 = 13/26 - 4/26 = 9/26.

Now, we can calculate how long it takes for the older machine to complete the remaining 9/26 of the work.

The older machine takes 15 hours to complete a one-half mile section of road, so it does 1/15 of the work in 1 hour.

To calculate how long it takes for the older machine to complete 9/26 of the work, we set up the following proportion:

(9/26) / 1 = 1 / x

Cross-multiplying, we get:

9x = 26

Dividing both sides by 9, we find:

x = 26/9 ≈ 2.9

Therefore, it takes the older machine approximately 2.9 hours to complete the job.

To solve this problem, we can determine the rate at which each machine paves the road and then use that information to find how long it will take the older machine to complete the job.

Let's first calculate the rate at which the new machine paves the road. We are given that the new machine takes 13 hours to complete a half-mile section of road. Therefore, its rate is 1/(13/2) = 2/13 miles per hour.

Next, we can determine how much of the road the new machine has paved before it developed the mechanical problem. We are told that it worked for 2 hours, so it has paved (2/13) * 1/2 = 1/13 of a mile.

Since the new machine quit working after 2 hours, the remaining section of road can be paved by the older machine. The older machine paves the same amount of road in 15 hours. Therefore, its rate is 1/(15/2) = 2/15 miles per hour.

To find how long it will take the older machine to complete the job, we need to determine the remaining distance to be paved. Since the new machine has already paved 1/13 of a mile, the remaining distance is 1/2 - 1/13 = 6/13 of a mile.

Finally, we can calculate the time it will take the older machine to complete the remaining distance by dividing the distance by its rate:
Time = Distance / Rate
Time = (6/13) / (2/15)
Time = (6/13) * (15/2)
Time = 90/26 ≈ 3.5 hours

Therefore, it will take the older machine approximately 3.5 hours to complete the job.