To solve this problem, let's carefully follow the mathematical steps needed.
Given:
1. Together, Sean and Colleen can clear the yard in 24 minutes.
2. Working alone, Sean takes 20 minutes longer than Colleen.
3. Let [tex]\( c \)[/tex] be the number of minutes it takes Colleen to finish the job alone.
4. The rational equation modeling this situation is:
[tex]\[
\frac{1}{c} + \frac{1}{c+20} = \frac{1}{24}
\][/tex]
Step-by-step solution:
1. Start with the given rational equation:
[tex]\[
\frac{1}{c} + \frac{1}{c+20} = \frac{1}{24}
\][/tex]
2. Multiply through by [tex]\( 24c(c+20) \)[/tex] to clear the denominators:
[tex]\[
24(c+20) + 24c = c(c+20)
\][/tex]
3. Simplify and expand both sides of the equation:
[tex]\[
24c + 480 + 24c = c^2 + 20c
\][/tex]
4. Combine like terms:
[tex]\[
48c + 480 = c^2 + 20c
\][/tex]
5. Move all terms to one side to form a quadratic equation:
[tex]\[
c^2 + 20c - 48c - 480 = 0
\][/tex]
Which simplifies to:
[tex]\[
c^2 - 28c - 480 = 0
\][/tex]
6. Solve the quadratic equation using the quadratic formula, [tex]\( c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -28 \)[/tex], and [tex]\( c = -480 \)[/tex]:
[tex]\[
c = \frac{28 \pm \sqrt{28^2 - 4 \cdot 1 \cdot (-480)}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
c = \frac{28 \pm \sqrt{784 + 1920}}{2}
\][/tex]
[tex]\[
c = \frac{28 \pm \sqrt{2704}}{2}
\][/tex]
[tex]\[
\sqrt{2704} = 52
\][/tex]
So:
[tex]\[
c = \frac{28 \pm 52}{2}
\][/tex]
7. This gives two potential solutions:
[tex]\[
c = \frac{28 + 52}{2} = \frac{80}{2} = 40
\][/tex]
[tex]\[
c = \frac{28 - 52}{2} = \frac{-24}{2} = -12
\][/tex]
8. Since [tex]\( c \)[/tex] represents time, it must be a positive value. Therefore, we discard [tex]\( -12 \)[/tex].
Hence, the time it would take Colleen to clear the yard alone is:
[tex]\[ \boxed{40} \][/tex]
So the correct answer is:
D. 40 minutes
