An ant starts at a foot of a staircase and crawls to the base of the fourth step. If each step is 8 inches tall and 10 inches wide, how far is the ant from where it began? The answer is 6(square root of 41) or 38.4. I have no clue in how to get to that answer.

the ant has gone 3 times the distances, or

30 inches along, and 24 inches up. Using your distance formula, the final displacement is

√(30^2+24^2)
= 6√(5^2+4^2)
= 6√41

Thank you, but how do you know the ant went 3 times the distances? When I was trying to figure it out, I thought that the ant would go 4 times the distance. Other than that, detailed and self-explanatory. Thank you

To find the distance the ant crawled, we need to calculate the length of its path along the staircase. Let's break down the problem step by step:

1. First, we need to find the diagonal distance covered by the ant. The ant crawls from the base of the staircase to the base of the fourth step. To calculate this diagonal distance, we can use the Pythagorean Theorem.

According to the Pythagorean Theorem, the square of the hypotenuse (diagonal distance) is equal to the sum of the squares of the other two sides. In this case, the two sides are the vertical height (8 inches) and the horizontal width (10 inches) of each step.

Applying the Pythagorean Theorem:
Hypotenuse² = Height² + Base²
Hypotenuse² = (8 inches)² + (10 inches)²
Hypotenuse² = 64 + 100
Hypotenuse² = 164

2. Now, we have the square of the hypotenuse. To find the actual length of the hypotenuse (diagonal distance), we need to take the square root of the value obtained in step 1.

Hypotenuse = √164 ≈ 12.81 inches

3. Since the ant crawls along the staircase, it doesn't crawl directly from the starting point to the end point. Instead, it moves along a zigzag path. Therefore, the distance covered by the ant is the sum of the vertical height (8 inches) and the diagonal distance (12.81 inches) obtained in step 2.

Distance covered = 8 inches + 12.81 inches ≈ 20.81 inches

Therefore, the ant is approximately 20.81 inches away from where it began.

It seems there was an error in the answer you provided (6√41 or 38.4). The correct answer should be around 20.81 inches.