One copy machine can run off copies in 170 minutes. A newer machine can do the same job in 119 minutes. How long does it take the two machines, working together, to make all of the necessary copies?
machine 1 does n copies/170 min
so n/170 copies/min
machine 2 does n/119 copies/min
so
n = (n/170 + n/119) t
1/t = 1/170+1/119 = .014286
so
t = 70 min
Well, it seems like these machines are in a bit of a race! With the older machine taking 170 minutes, it's definitely lagging behind. Meanwhile, the newer machine is speeding ahead at 119 minutes.
Now, when these two machines decide to team up, it's like having a slowpoke and a speed demon holding hands. They're going to find a pace somewhere in the middle.
To figure out how long it takes for them to make all the necessary copies together, we can use the concept of rates. The older machine can make 1/170th of the copies per minute, while the newer machine can make 1/119th of the copies per minute.
When we add these rates together, we get (1/170 + 1/119) copies per minute. To simplify things, we can find a common denominator, which is 170 * 119, equal to 20230.
So, the combined rate is (119 + 170) / 20230 copies per minute. Simplifying that further, we get 289 / 20230 copies per minute.
To find the time it takes them to make all the necessary copies, we need to take the reciprocal of the rate:
1 / (289 / 20230) minutes per copy.
Crunching the numbers, that turns out to be approximately 70.16 minutes.
So, these two machines, with their contrasting personalities, will take about 70.16 minutes to make all of the necessary copies when they work together. They may have their differences, but they create quite the copy-making duo!
To find the time it takes for the two machines to make all the necessary copies, we can use the concept of their combined work rate.
Let's assume that the amount of work required is 1 job (in this case, making all the necessary copies).
The work rate of the first machine is 1 job per 170 minutes, and the work rate of the second machine is 1 job per 119 minutes.
To determine their combined work rate, we add their individual work rates:
Combined work rate = Work rate of machine 1 + Work rate of machine 2
Combined work rate = 1/170 + 1/119
Next, we find the reciprocal of the combined work rate to get the time it takes for both machines to complete 1 job:
Time taken = 1 / Combined work rate
Time taken = 1 / (1/170 + 1/119)
Calculating the expression inside the parentheses:
Time taken = 1 / (0.005882 + 0.008403)
Time taken = 1 / 0.014285
Time taken ≈ 70 minutes
Therefore, it would take the two machines, working together, approximately 70 minutes to make all the necessary copies.
To find out how long it takes for the two machines to make all the necessary copies when working together, we can use the concept of work done per unit time.
Let's say the amount of work required is represented by a 1. This is because the problem doesn't provide specific information on the number of copies needed, but we can assume that the work required is a unit amount for simplicity.
The work done by the first machine in one minute can be represented as 1/170 since it takes 170 minutes for the machine to complete the entire job.
Similarly, the work done by the newer machine in one minute can be represented as 1/119 since it takes 119 minutes for the newer machine to complete the entire job.
Now, when the two machines work together, their work rates are additive. So, the combined work done by the two machines in one minute would be:
1/170 + 1/119 = (119 + 170)/(170 * 119) = 289/20230
Now, to find the time it takes for the two machines working together to make all the necessary copies, we can set up a proportion:
(289/20230) / x = 1/1
Where x represents the time in minutes.
Cross-multiplying the proportion, we get:
(289/20230) * 1 = 1 * x
Simplifying:
289/20230 = x
x ≈ 0.0143
Therefore, it takes approximately 0.0143 minutes (or approximately 0.858 seconds) for the two machines to make all the necessary copies when working together.
rate of old machine = 1/170
rate of new machine = 1/119
combined rate = 1/170 + 1/119
time at combined rate = 1/(1/170+1/119)
= 1/(1/70) = 70 minutes