Points A and B are on the top and bottom edges of a cylindrical roll of paper of height 8 and circumference 12. A and B are diagonally opposite each other. The paper is cut along line C and opened out. How far apart are A and B on the flat surface?
To find the distance between points A and B on the flat surface after the paper is cut and opened out, we can use basic geometry.
First, let's visualize the situation. When the paper is cut along line C and opened out, it will become a rectangle with a length equal to the circumference of the cylindrical roll (12 units) and a width equal to the height of the cylindrical roll (8 units).
Now, we want to find the distance between points A and B on this rectangle.
Since A and B are diagonally opposite each other on the cylindrical roll, they will also be diagonally opposite each other on the rectangle. This means the distance between them creates a diagonal of the rectangle.
To find the length of this diagonal, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In our case, the width and length of the rectangle form the two sides of the triangle, and the distance between A and B is the hypotenuse.
Let's label the width of the rectangle as 'w', the length of the rectangle as 'l', and the distance between A and B on the flat surface as 'd'.
We know that w (width) = 8 units and l (length) = 12 units.
Using the Pythagorean theorem, we can calculate the value of 'd':
d^2 = w^2 + l^2
d^2 = 8^2 + 12^2
d^2 = 64 + 144
d^2 = 208
To find 'd', we need to take the square root of both sides of the equation:
d = √208
d ≈ 14.42
Therefore, the distance between points A and B on the flat surface is approximately 14.42 units.