The stairway is 10 ft high, with each step rising 0.5 ft upward and extending 1 ft forward.
Number of steps = total height / height per step = 10 ft / 0.5 ft = 20 steps.
For each step, the ant travels the hypotenuse of a right triangle with legs 0.5 ft (up) and 1 ft (forward).
Distance per step = √(0.5² + 1²) = √(0.25 + 1) = √1.25 = √(5/4) = √5 / 2 ft.
Total distance = 20 × (√5 / 2) = 10√5 ft.
√5 ≈ 2.236, so 10 × 2.236 = 22.36 ft.
None of the options match (30, 33, 10, 29 ft). The options seem off for the hypotenuse path.
Perhaps the ant is crawling along the surface, but maybe they mean the total along the steps without hypotenuse—up + forward per step.
Per step: 0.5 up + 1 forward = 1.5 ft.
20 steps × 1.5 = 30 ft.
That matches option A.
If it's a straight ramp, the slant height would be √(10² + (20×1)²) = √(100 + 400) = √500 = 10√5 ≈ 22.36 ft again, same as before.
But since it's stepped, the surface path is the sum of the hypotenuses of each step, which is also 10√5 ft.
The question says "each step accounts for half a foot upward and one foot forward." This describes the riser and tread of each step.
For an ant crawling on the surface of the stairs, it would go along the tread (1 ft forward) then up the riser (0.5 ft), or depending on path, but typically the shortest path on surface is along the hypotenuse if it cuts corners, but no—for discrete steps, if it follows the profile, it's tread + riser per step.
The standard "ant on stairs" problem usually has the ant crawling along the surface, so total distance is number of steps × (tread + riser).
Here: 20 × (1 + 0.5) = 30 ft.
And 30 ft is option A.
If the ant goes straight up the diagonal of the entire staircase as if it were a ramp, it's shorter: 10√5 ≈22.36 ft, but that's not an option.
The options are 30, 33, 10, 29—likely the intended answer is 30 ft, treating it as sum of vertical and horizontal per step.
Some might mistakenly do 20 up + 20 forward? No, height is 10, forward total 20 ft, hypotenuse 22.36 or sum 30.
Or perhaps 10/0.5=20 steps, but then 20×1 forward=20, plus height 10=30 again.
Yes.
I think the question intends the ant to follow the steps: total vertical distance traveled is 10 ft, but no—the distance traveled is the path length.
But in multiple choice, and common such problems, it's often the unfolded path: sum of all treads and risers = total horizontal + total vertical = 20 ft + 10 ft = 30 ft.
Yes, that's a common trick question—people might think hypotenuse, but for following the stairs surface without cutting across, it's 30 ft.
But actually, an ant could take a straighter path across multiple steps diagonally, but the problem likely assumes following the stair profile.
Given the options, 30 ft is there, while √(10²+20²) isn't.
So the answer should be A) 30 ft.
To confirm precisely:
Total rise: 10 ft
Total run: 20 steps × 1 ft = 20 ft
If ant crawls along each riser and tread separately: distance = total rise + total run = 30 ft.
Yes, that's the standard interpretation for such problems when options match.
The hypotenuse per step way: each step's surface if considering the ant going diagonally on each step face, but since steps are separate, but anyway, if the step has a flat tread and vertical riser, the shortest on one step is hypotenuse, total still 10√5, but again not matching options.
But many such exam questions expect 30 ft.
