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Sagot :
To solve the problem, let's start by setting up the equation based on the information given in the problem.
We are given that:
- The total time taken by Jay and Kevin together to clear the driveway is 14 minutes.
- Jay alone would take [tex]\( x \)[/tex] minutes to clear the driveway.
- Kevin alone would take [tex]\( x + 6 \)[/tex] minutes to clear the driveway (as it takes him 6 minutes more than Jay).
The equation derived from these facts is:
[tex]\[ \frac{1}{x} + \frac{1}{x + 6} = \frac{1}{14} \][/tex]
We need to solve this equation to find the value of [tex]\( x \)[/tex], the time it takes Jay to clear the driveway alone.
1. Combine the fractions on the left-hand side over a common denominator:
[tex]\[ \frac{x + 6 + x}{x(x + 6)} = \frac{1}{14} \][/tex]
This simplifies to:
[tex]\[ \frac{2x + 6}{x(x + 6)} = \frac{1}{14} \][/tex]
2. Cross-multiply to eliminate the denominators:
[tex]\[ 14(2x + 6) = x(x + 6) \][/tex]
3. Distribute and rearrange the equation into standard quadratic form:
[tex]\[ 28x + 84 = x^2 + 6x \][/tex]
[tex]\[ x^2 + 6x - 28x - 84 = 0 \][/tex]
[tex]\[ x^2 - 22x - 84 = 0 \][/tex]
4. Factor or use the quadratic formula to solve for [tex]\( x \)[/tex]. Factoring this particular quadratic equation directly might be challenging, so we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -22 \)[/tex], and [tex]\( c = -84 \)[/tex]:
[tex]\[ x = \frac{22 \pm \sqrt{22^2 - 4 \cdot 1 \cdot (-84)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{22 \pm \sqrt{484 + 336}}{2} \][/tex]
[tex]\[ x = \frac{22 \pm \sqrt{820}}{2} \][/tex]
[tex]\[ x = \frac{22 \pm 28.64}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{22 + 28.64}{2} \approx 25.32 \][/tex]
[tex]\[ x = \frac{22 - 28.64}{2} \approx -3.32 \][/tex]
Given the context of the problem, [tex]\( x \)[/tex] must be a positive value, so we discard the negative solution.
Therefore, the positive, practical solution is:
[tex]\[ x \approx 25.32 \text{ minutes} \][/tex]
Hence, the closest correct answer from the given options is not listed precisely. However, we conclude from our calculation that Jay would take approximately 25.32 minutes to clear the driveway alone.
From the closest available options, it is prudent to notice that the question might contain an error in the choices, but numerically, Jay takes around 25.32 minutes. The question's options should be revisited for accuracy.
Properly, none of A, B, C, or D are valid, and the answer approximately should mention 25.32 minutes or closely 25 minutes if approximated.
We are given that:
- The total time taken by Jay and Kevin together to clear the driveway is 14 minutes.
- Jay alone would take [tex]\( x \)[/tex] minutes to clear the driveway.
- Kevin alone would take [tex]\( x + 6 \)[/tex] minutes to clear the driveway (as it takes him 6 minutes more than Jay).
The equation derived from these facts is:
[tex]\[ \frac{1}{x} + \frac{1}{x + 6} = \frac{1}{14} \][/tex]
We need to solve this equation to find the value of [tex]\( x \)[/tex], the time it takes Jay to clear the driveway alone.
1. Combine the fractions on the left-hand side over a common denominator:
[tex]\[ \frac{x + 6 + x}{x(x + 6)} = \frac{1}{14} \][/tex]
This simplifies to:
[tex]\[ \frac{2x + 6}{x(x + 6)} = \frac{1}{14} \][/tex]
2. Cross-multiply to eliminate the denominators:
[tex]\[ 14(2x + 6) = x(x + 6) \][/tex]
3. Distribute and rearrange the equation into standard quadratic form:
[tex]\[ 28x + 84 = x^2 + 6x \][/tex]
[tex]\[ x^2 + 6x - 28x - 84 = 0 \][/tex]
[tex]\[ x^2 - 22x - 84 = 0 \][/tex]
4. Factor or use the quadratic formula to solve for [tex]\( x \)[/tex]. Factoring this particular quadratic equation directly might be challenging, so we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -22 \)[/tex], and [tex]\( c = -84 \)[/tex]:
[tex]\[ x = \frac{22 \pm \sqrt{22^2 - 4 \cdot 1 \cdot (-84)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{22 \pm \sqrt{484 + 336}}{2} \][/tex]
[tex]\[ x = \frac{22 \pm \sqrt{820}}{2} \][/tex]
[tex]\[ x = \frac{22 \pm 28.64}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{22 + 28.64}{2} \approx 25.32 \][/tex]
[tex]\[ x = \frac{22 - 28.64}{2} \approx -3.32 \][/tex]
Given the context of the problem, [tex]\( x \)[/tex] must be a positive value, so we discard the negative solution.
Therefore, the positive, practical solution is:
[tex]\[ x \approx 25.32 \text{ minutes} \][/tex]
Hence, the closest correct answer from the given options is not listed precisely. However, we conclude from our calculation that Jay would take approximately 25.32 minutes to clear the driveway alone.
From the closest available options, it is prudent to notice that the question might contain an error in the choices, but numerically, Jay takes around 25.32 minutes. The question's options should be revisited for accuracy.
Properly, none of A, B, C, or D are valid, and the answer approximately should mention 25.32 minutes or closely 25 minutes if approximated.
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