Final answer:
Beth takes approximately 6.67 hours to complete the roof alone, while Dave takes about 13.34 hours since he works at half the speed. By setting up an equation based on their combined work rate, the individual times can be found.
Step-by-step explanation:
To find out how long it takes for Dave and Beth to complete a roof alone, we set up a scenario where Dave takes twice as long as Beth to complete a roof. Let's define the time it takes for Beth to complete the roof alone as 'x' hours. Since Dave takes twice as long, he takes '2x' hours. When they work together, they can complete the roof in 10 hours.
Together, in one hour, Beth completes 1/x of the work and Dave completes 1/(2x) of the work. Their combined work for one hour is 1/x + 1/(2x), which equals (3/2x). This rate times 10 hours should equal 1 job done, so we have the equation (3/2x) * 10 = 1. Solving for x gives us x = 10/(3/2), which simplifies to x = 20/3 or approximately 6.67 hours for Beth. Consequently, it will take Dave 2*6.67 = 13.34 hours to complete the roof alone.