Working together, Sandy and Carol can pack a storage unit in 4 hours. If they work alone, it takes Sandy 6 hours longer than it takes Carol. How long would it take Carol to pack the storage unit alone?
If Carol takes x hours, then
1/x + 1/(x+6) = 1/4
To solve this problem, we can set up a system of equations. Let's use the following variables:
Let x be the amount of time it takes Carol to pack the storage unit alone (in hours).
Then, since Sandy takes 6 hours longer than Carol, it would take Sandy (x + 6) hours to pack the storage unit alone.
Now, we can use the concept of work rates. Sandy and Carol together complete 1 storage unit in 4 hours, so their combined work rate is 1/4 of a storage unit per hour.
Sandy's work rate is 1/(x+6) of a storage unit per hour, and Carol's work rate is 1/x of a storage unit per hour.
Since work rate is inversely proportional to the time it takes to complete a task, we can set up the equation:
1/(x+6) + 1/x = 1/4
To solve this equation, we can find a common denominator and then combine the fractions:
(x+x+6)/(x(x+6)) = 1/4
Simplifying the left side of the equation:
(2x+6)/(x^2+6x) = 1/4
Cross-multiplying to eliminate fractions:
4(2x+6) = x^2+6x
Expanding and simplifying:
8x + 24 = x^2 + 6x
Rearranging the equation:
x^2 - 2x - 24 = 0
Now, we can factor or use the quadratic formula to solve for x. Factoring gives us:
(x - 6)(x + 4) = 0
So, x = 6 or x = -4. Since we are dealing with time, we discard the negative solution.
Therefore, it would take Carol 6 hours to pack the storage unit alone.