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Answer:
The time it takes the first kitten to unwind the balls of yarn alone is 15 minutes
The time it takes the second kitten to unwind the balls of yarn alone is 10 minutes
Step-by-step explanation:
The time it takes the two kitten to unwind the balls of yarn = 9 minutes faster than the first kitten working alone
The time it takes the two kitten to unwind the balls of yarn = 4 minutes faster than the second kitten working alone
Let 'x' represent the time it takes the first kitten to unwind the balls of yarn alone, let 'y' represent the time it takes the second kitten to unwind the balls of yarn alone, and let 't' represent the time it takes the two kitten to unwind the balls of yarn we have;
x = t + 9
y = t + 4
The rate of working, per ball of yarn, of the first kitten = 1/x = 1/(t + 9)
The rate of working, per ball of yarn, of the second kitten = 1/y = 1/(t + 4)
The amount of work each kitten does in the time 't', added together to unwind 1 ball of yarn is given as follows;
t/(t + 9) + t/(t + 4) = 1
(t·(t +4) + t·(t + 9))/(t + 4)×(t + 9) = 1
Using a graphing calculator, we have;
(2·t²+13·t)/(t² + 13·t + 36) = 1
∴ (2·t² + 13·t) = (t² + 13·t + 36)
(2·t² + 13·t) - (t² + 13·t + 36) = 0
t² - 36 = 0
t = √36 = 6
t = 6
x = t + 9 = 6 + 9 = 15
The time it takes the first kitten to unwind the balls of yarn alone, x = 15 minutes
y = t + 4 = 6 + 4 = 10
The time it takes the second kitten to unwind the balls of yarn alone, y = 10 minutes