An ant walks along a helical path on a solid cylindrical object of diameter 5 cm and height 5 cm. It starts walking from point X, goes around the object, and reaches point Y, which is vertically below point X as shown in the figure. How much distance did the ant walk?
Show Hint
To calculate the distance of a helical path, use the Pythagorean theorem where the height and circumference form the two legs of the right triangle.
Step 1: Understanding the path.
The ant walks along a helical path. The helical distance can be found using the Pythagorean theorem. The vertical distance is the height of the cylinder, and the horizontal distance is the circumference of the circle formed by the cylindrical object.
Step 2: Calculating the horizontal distance.
The diameter of the cylinder is given as 5 cm, so the radius \( r \) is:
\[
r = \frac{5}{2} = 2.5 \, \text{cm}
\]
The circumference \( C \) of the circle is:
\[
C = 2\pi r = 2\pi(2.5) = 5\pi \, \text{cm}
\]
Step 3: Applying the Pythagorean theorem.
The ant walks a helical path, so the total distance \( D \) it travels is the hypotenuse of a right triangle, where one leg is the height of the cylinder (5 cm) and the other leg is the circumference of the cylinder (\( 5\pi \) cm). Using the Pythagorean theorem:
\[
D = \sqrt{(5)^2 + (5\pi)^2} = \sqrt{25 + 25\pi^2}
\]
\[
\boxed{\sqrt{25 + 25\pi^2} \, \text{cm}}
\]
