Final answer:
The correct option is D). To determine Geneva's and Miles's individual mowing times, we use their combined mowing rate to set up and solve an equation. Geneva's mowing time is found to be 90 minutes and Miles's is 120 minutes.
Step-by-step explanation:
The question involves finding the individual mowing times for Geneva and Miles, given that they can mow a lawn together in 2 hours and that Miles, if working alone, would take an additional 30 minutes longer than Geneva to mow the lawn. To solve this, we can let Geneva's mowing time be x hours and therefore, Miles's mowing time would be x + 0.5 hours. The combined rate of work can be represented by the equation 1/x + 1/(x + 0.5) = 1/2 since together they take 2 hours. Solving for x gives us the individual mowing times.
Solving the equation: 1/x + 1/(x + 0.5) = 1/2, we first find a common denominator and multiply through to get 2(x + 0.5) + 2x = x(x + 0.5), giving us a quadratic equation. Simplifying the equation, we get 2x + 1 + 2x = x^2 + 0.5x, which simplifies to x^2 - 3.5x - 1 = 0. Solving this quadratic equation yields x = 3/2 (or 1.5 hours) as the only positive solution for Geneva's mowing time and for Miles, it would be x + 0.5 = 2 hours. Therefore, Geneva takes 90 minutes and Miles takes 120 minutes individually to mow the lawn.