In an assembly line of a factory, which produces networking switching hubs, two robots are utilized. The first one (A) is able to produce 40 microchips per hour, whereas the second one (B) produces 30 microchips per hour. For a total production of 400 microchips, the two robots operated for 12 hours. How many hours did each robot work?
(1) clearly define your variables: state what they represent and what their relations are in terms of the information given in the problem statement,
(2) setup your equations,
(3) solve the system,
(4) check/validate your solutions and
(5) summarize your findings in a sentence that refers to what the problem ask
1) Variables:
Let's define the variables:
A = number of hours Robot A worked
B = number of hours Robot B worked
2) Equations:
From the problem statement, we know that Robot A can produce 40 microchips per hour, while Robot B can produce 30 microchips per hour. So, the total number of microchips produced by Robot A is given by A * 40 and by Robot B is given by B * 30.
The total production of microchips is given as 400, so we have the equation: A * 40 + B * 30 = 400
We are also given that the two robots worked for a total of 12 hours, so we have the equation: A + B = 12
3) Solving the system of equations:
We have two equations with two unknowns, A and B. We can solve this system using any method of our choice. In this case, let's solve it using substitution.
First, we can solve the second equation for A:
A + B = 12
A = 12 - B
Now substitute this value of A in the first equation:
(12 - B) * 40 + B * 30 = 400
480 - 40B + 30B = 400
-10B = -80
B = 8
Substituting the value of B back into the second equation:
A + 8 = 12
A = 4
4) Check/Validate:
To validate our solutions, let's substitute the values of A = 4 and B = 8 into the original equations:
Equation 1: A * 40 + B * 30 = 400
4 * 40 + 8 * 30 = 400
160 + 240 = 400
400 = 400 (True)
Equation 2: A + B = 12
4 + 8 = 12
12 = 12 (True)
Both equations hold true, so our solutions are valid.
5) Summary:
Robot A worked for 4 hours, while Robot B worked for 8 hours to produce a total of 400 microchips.