A pirate starts at point A, takes 54 paces due west, turns and takes 34 paces due south, then turns again and takes 12 paces due East to find a buried treasure. How many paces from the buried treasure is point A?
p^2 = (54 - 12)^2 + 34^2
X = -54 + 12 = -42 paces.
Y = -34 paces.
d^2 = x^2 + y^2 = (-42)^2 + (-34)^2 =
d = ?
To find the distance from point A to the buried treasure, we can use the concepts of vectors and Pythagorean theorem.
Let's break down the pirate's movements into two perpendicular components: north-south (vertical) and east-west (horizontal). The problem states that the pirate took 54 paces west, 34 paces south, and then 12 paces east.
First, we need to calculate the total westward and eastward movement. Since the pirate walked 54 paces west and then 12 paces east, the net horizontal movement is 54 - 12 = 42 paces west.
Next, we calculate the total southward and northward movement. Since the pirate walked 34 paces south and there is no mention of any northward movement, the net vertical movement is 34 paces south.
Now, we have a right-angled triangle with the horizontal distance (42 paces) as the base and the vertical distance (34 paces) as the height. We can use the Pythagorean theorem, which states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):
c^2 = a^2 + b^2
In this case, the hypotenuse length represents the distance from point A to the buried treasure, the base length represents the horizontal distance, and the height length represents the vertical distance.
Applying the Pythagorean theorem, we have:
Distance^2 = (42 paces)^2 + (34 paces)^2
Distance^2 = 1764 paces^2 + 1156 paces^2
Distance^2 = 2920 paces^2
Taking the square root of both sides, we get:
Distance = √(2920 paces^2)
Calculating the square root, we find:
Distance ≈ 54.03 paces
Therefore, the distance from point A to the buried treasure is approximately 54.03 paces.