A factory kept increasing it's output by the same percentage every year .find the percentage if it is known that the output is doubled in the last two years.
2 = [1 + (p / 100)]^2
√2 = 1 + (p / 100)
To find the percentage increase each year, we need to work backwards from the final output. Let's assume the initial output of the factory is "x".
After the first year, the output would be increased by a certain percentage, let's say "P1". So, after one year, the output becomes "x + (P1/100)x".
Now, after the second year, the output is doubled, so it becomes 2x.
Using this information, we can set up an equation:
2x = (x + (P1/100)x) + (x + (P1/100)x) * (P1/100)
Simplifying this equation, we get:
2 = 1 + (1 + P1/100) * (1 + P1/100)
2 = 1 + (1 + 2P1/100 + P1^2/10000)
2 = 1 + 1 + 2P1/100 + P1^2/10000
Now, let's simplify and solve for P1.
Multiplying throughout by 10000, we get:
20000 = 10000 + 10000P1 + P1^2
Rearranging the equation, we get:
P1^2 + 10000P1 + 10000 = 20000
P1^2 + 10000P1 - 10000 = 0
Now we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula.
Once we solve for P1, we will have the percentage increase each year.