An automated manufacturing plant uses robots to manufacture products. A generation-I robot working alone can manufacture a product in 30 hours whereas a generation-II robot working alone can manufacture the same product in 20 hours. If the manager of the plant wants the product manufacturing time to be greater than equal to 5 hours with at least 1 robot each of both generations working together, how many possible combinations of the number of robots of each generation is possible?a)4b)5c)6d)7e)8Correct answer is option 'E'. Can you explain this answer?

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An automated manufacturing plant uses robots to manufacture products. A generation-I robot working alone can manufacture a product in 30 hours whereas a generation-II robot working alone can manufacture the same product in 20 hours. If the manager of the plant wants the product manufacturing time to be greater than equal to 5 hours with at least 1 robot each of both generations working together, how many possible combinations of the number of robots of each generation is possible?

a)
4
b)
5
c)
6
d)
7
e)
8

Correct answer is option 'E'. Can you explain this answer?

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An automated manufacturing plant uses robots to manufacture products. ...

Given:

Generation-I Robot
- Time taken to manufacture a product = 30 hours
Generation-II Robot
- Time taken to manufacture the product = 20 hours
To Find: Possible combinations of number of generation-I and II robots such that the product manufacturing time ≥ 5 hours
Let the number of generation-I robots be x and number of generation-II robots be y

Approach:

We need to find the possible combination of the values of (x, y).
As we are given that the time taken for x generation-I and y generation-II robots should be greater than equal to 5 hours, we will formula an inequality in time taken for x generation robots and y generation-II robots to evaluate the possible combination of (x, y)
As Work = Rate * Time, for finding the time taken by the robots, we would need to find the work done and the rate
Number of products to be produced = 1
Now, we know that rate of production of n robots = n * rate of production of 1 robot.
So, once we find the rate of production of 1 robot for each generation, we can find the rate of production of x generation-I robots and y generation-II robots respectively.
As we are given the time taken by 1 generation-I robot and 1 generation-II robot to produce 1 product each, we can find the rate of production for robots of both generations.

Working out:

Generation-I
a. Work Done = 1 product
b. Time taken = 30 hours
c. Rate of manufacturing products = products per hour

Generation-II
a. Work Done = 1 product
b. Time taken = 20 hours
c. Rate of manufacturing products = 1/20 products per hour
3. Let’s assume there are x generation-I robots and y generation-II robots working together to produce 1 unit of product
a. So, work done = 1 product

b. Rate of x generation-I robots =

c. Rate of y generation-II robots =

4. As the time taken ≥ 5 hours, we can write

5. So, the possible combinations of {x, y} can be:
a. If y = 1, x = {1,2,3,4} – 4 options
b. If y = 2, x = {1,2,3} – 3 options
c. If x = 1, y = {1,2,3} – 3 options
d. However {x, y} = {1,1} and {1,2} is repeated.

6. So, we have a total of 10 -2 = 8 possible options
a. Please note that we have not considered x, y = 0 as we are given that at-least 1 robot of each generation should be working

Answer : E

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