A stairway 10 ft. high is such that each step accounts for half a foot upward and one foot forward. What distance will an ant travel if it starts from ground level to reach the top of the stairway?

### Concept & Logic A staircase consists of alternating vertical rises and horizontal treads. Reaching the "top" of a staircase logically means ascending the final riser to land on the upper floor level, rendering an additional forward tread unnecessary to satisfy the height completion. ### Step-by-Step Solution * **Given:** * Total stairway height = 10 ft. * Single step vertical rise = 0.5 ft. * Single step horizontal forward (tread) = 1 ft. * **Calculation:** First, calculate the total number of upward steps (rises) needed to reach 10 ft: $$\text{Total steps} = \frac{10}{0.5} = 20 \text{ steps}$$ The ant must travel upward for every step: $$\text{Total vertical distance} = 20 \times 0.5 = 10 \text{ ft}$$ For $n$ steps reaching a top landing, there are only $n-1$ intermediate forward treads. The 20th riser puts the ant securely on the top floor. $$\text{Total horizontal distance} = 19 \times 1 = 19 \text{ ft}$$ Sum the vertical and horizontal distances traveled: $$\text{Total distance} = 10 + 19 = 29 \text{ ft}$$ ### Exam Strategy & Shortcut Memorize the geometry of stairs for an entity climbing $N$ steps: $$\text{Total Distance} = (N \times \text{rise}) + ((N-1) \times \text{tread})$$ Here $N=20$. Distance = $10 + 19 = 29$. You can do this visually in your head in seconds. ### Common Pitfall Assuming there are equal numbers of vertical rises and horizontal treads (20 of each). This leads to calculating $10 + 20 = 30$ ft, tricking students into selecting Option (a). Always remember that the last step up does not require a subsequent tread to be considered "at the top." ### Final Answer **Therefore, the correct answer is 29 ft.**