Find the length of the longest rod that will fit inside a cylinder with a diameter of 2020 cm and a height of 7575 cm. Answer correct to 11 decimal place.

The longest rod that can fit inside a cylinder is the one that goes from one bottom corner to the top corner diagonally opposite, which is the same as the length of the space diagonal of the cylinder.

The cylinder can be thought of as a right circular cone, and the longest rod is the hypotenuse of the right triangle formed by the height of the cylinder and the diameter of the base.

We can use the Pythagorean theorem to find the length of the longest rod. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). In mathematical terms, this is expressed as:

c² = a² + b²

In this case, the height of the cylinder (7575 cm) is one side of the right triangle (a), and the diameter of the base (2020 cm) is the other side (b). We can substitute these values into the Pythagorean theorem to find the length of the longest rod (c):

c² = (7575 cm)² + (2020 cm)² c² = 57305625 cm² + 4080400 cm² c² = 61386025 cm²

Taking the square root of both sides to solve for c gives:

c = sqrt(61386025 cm²) c = 7835.000000000 cm

So, the length of the longest rod that will fit inside the cylinder is approximately 7835.000000000 cm.