Final answer:
By solving the given rational equation, we find that Colleen would take 12 minutes alone to clear the yard of leaves, which is option A.
Step-by-step explanation:
To determine how long it would take Colleen alone to clear the yard of leaves, we need to solve the rational equation given by 1/c + 1/(c + 20) = 1/24. Since Sean would take 20 minutes longer than Colleen, if it takes Colleen c minutes, then it takes Sean c + 20 minutes to do the job alone. The reciprocal of the time taken by each gives us their rate of work per minute.
Combining their rates gives us the rate at which they work together, which is 1/24 yards per minute. We set up the equation:
1/c + 1/(c + 20) = 1/24.
By finding a common denominator and equating the numerators, we can solve for c:
c(c + 20) + c(24) = (c + 20)(24).
Expanding and simplifying, we find:
c2 + 20c + 24c = 24c + 480,
c2 + 44c = 480.
Bringing all terms to one side gives us a quadratic equation:
c2 + 44c - 480 = 0.
Factoring this quadratic, we get:
(c - 12)(c + 40) = 0,
which gives us two possible solutions for c, c = 12 or c = -40. Since time cannot be negative, we discard c = -40. Thus, the only feasible solution is c = 12 minutes.
Therefore, Colleen would take 12 minutes to clear the yard of leaves working alone, which corresponds with option A.