PLEASE HELP! D: THANKS SO MUCH! An ant wants to travel from the corner of a 1x2x4 brick to the opposite corner. What is the length of the shortest path it can take? Hint: You have to check three different paths.

Step-by-step explanation:

Assuming the ant can only walk along the surface of the brick and not tunnel through it using its formic acid to dissolve a pathway,  a short path would be to cut diagonally across the 2 x 4 face and then walk along the 1 unit edge

d = 1 + √(2² + 4²) = 1 + √20 ≈ 5.47 units

the other possibilities are

d = 2 + √(1² + 4²) = 2 + √17 ≈ 6.12 units

d = 4 + √(1² + 2²) = 4 + √5 = 6.23 units

Actually there is a slightly shorter path. It will be two diagonals along the two largest surfaces.

Assume x is the distance along a 4 inch side from the corner opposite corner of initiation.

The distance traveled will be

d = √((4 - x)² + 2²) + √(x² + 1²)

d = √((x² - 8x + 16) + 4) + √(x² + 1²)

d = √(x² - 8x + 20) + √(x² + 1)

the minimum distance will occur when the derivative of the distance is set to zero and solved for x

d' = 2x - 8 / 2√(x² - 8x + 20) + x / √(x² + 1)

Which has a zero at x = 1⅓

so minimum travel distance is

d = √((1⅓)² - 8(1⅓) + 20) + √((1⅓)² + 1) = 5 units

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