Diane Gray and Lucy Hoang together can prepare a crawfish boil for a lagre party in 4 hours. Lucy alone can complete the job in 2 hours less time than Diane alone. Find the time in which each person can prepare the crawfish boil alone. Round to the nearest tenth of an hour.
If Diane takes x hours,
1/x + 1/(x-2) = 1/4
Let's assume that Diane can prepare the crawfish boil alone in x hours.
According to the given information, Lucy can complete the job in 2 hours less than Diane alone. Therefore, Lucy can do the job in (x-2) hours.
Now, we know that Diane and Lucy together can prepare the crawfish boil in 4 hours. So, their combined work rate is 1/4 of the job per hour.
To find their individual work rates, we can use the following equation:
1/x + 1/(x-2) = 1/4
Now, let's solve this equation to find the values of x and x-2:
Multiply both sides of the equation by 4x(x-2) to eliminate the denominators:
4(x-2) + 4x = x(x-2)
Expand the equation:
4x - 8 + 4x = x^2 - 2x
Combine like terms:
8x - 8 = x^2 - 2x
Rearrange the equation to put it in quadratic form:
x^2 - 2x - 8x + 8 - 0 = 0
Simplify the equation:
x^2 - 10x + 8 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 1, b = -10, and c = 8.
Plug in the values:
x = (-(-10) ± √((-10)^2 - 4(1)(8)))/(2(1))
Simplify:
x = (10 ± √(100 - 32))/2
x = (10 ± √68)/2
x = (10 ± 2√17)/2
x = 5 ± √17
Now, we have two possible solutions for x: x = 5 + √17 and x = 5 - √17. These represent the times in which Diane can prepare the crawfish boil alone.
Rounded to the nearest tenth of an hour, Diane can prepare the crawfish boil alone in approximately 7.12 hours (5 + √17) and Lucy can prepare it alone in approximately 5.12 hours (7.12 - 2).