Select the correct answer. Sean and Colleen are raking leaves in their yard. Working together, they can clear the yard of leaves in 24 minutes. Working alone, it would take Sean 20 minutes longer to clear the yard than it would take Colleen working alone. When \( c \) is the number of minutes it would take Colleen to.finish the job when working alone, the situation is modeled by this rational equation: \( \frac{1}{e}+\frac{1}{c+20}=\frac{1}{24} \). How long would it take Colleen alone to clear the yard of leaves? A. 12 minutes B. 14 minutes C. 28 minutes D. 40 minutes
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To solve the rational equation \( \frac{1}{e} + \frac{1}{c+20} = \frac{1}{24} \), you can start by letting \( c \) be the time Colleen takes alone. Since Sean takes 20 minutes longer, we have \( e = c + 20 \). Substituting this into the equation, we get: \[ \frac{1}{c} + \frac{1}{(c + 20)} = \frac{1}{24}. \] To find a common denominator, you can multiply both sides by \( 24c(c + 20) \): \[ 24(c + 20) + 24c = c(c + 20). \] This simplifies to: \[ 24c + 480 + 24c = c^2 + 20c. \] Combining like terms and moving everything to one side gives: \[ 0 = c^2 - 28c - 480. \] Now, applying the quadratic formula: \[ c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{28 \pm \sqrt{784 + 1920}}{2}. \] Calculating further, the discriminant simplifies to: \[ c = \frac{28 \pm \sqrt{2704}}{2} = \frac{28 \pm 52}{2}. \] This gives \( c = 40 \) or \( c = -12 \) (not possible, since time can't be negative). Thus, Colleen would take **40 minutes** to clear the yard alone, corresponding to option D.
