From a circular sheet of paper of radius 25 cm, a sector area 4% is removed. If the remaining part is used to make a conical surface, then find the ratio of the radius and height of the cone.

Sure, let's solve this step by step:

Step 1: Calculate the circumference of the original circle. The circumference of a circle is given by the formula 2πr, where r is the radius of the circle. Here, the radius is 25 cm, so the circumference is 2π*25 = 50π cm.

Step 2: Calculate the length of the sector that was removed. The sector that was removed is 4% of the total circle. So, the length of the sector that was removed is 4% of the total circumference, which is 0.04*50π = 2π cm.

Step 3: Calculate the circumference of the remaining circle. The remaining circle's circumference is the original circumference minus the length of the sector that was removed. So, the remaining circumference is 50π - 2π = 48π cm.

Step 4: The remaining part is used to make a conical surface. The circumference of the base of the cone is equal to the circumference of the remaining circle. So, the radius of the base of the cone is the remaining circumference divided by 2π, which is 48π/2π = 24 cm.

Step 5: The slant height of the cone is equal to the radius of the original circle, which is 25 cm.

Step 6: The height of the cone can be found using the Pythagorean theorem, since the radius, slant height, and height form a right triangle in the cone. The Pythagorean theorem is a² + b² = c², where c is the hypotenuse (the slant height), a is one leg (the radius), and b is the other leg (the height). Solving for b, we get b = sqrt(c² - a²) = sqrt(25² - 24²) = sqrt(625 - 576) = sqrt(49) = 7 cm.

Step 7: The ratio of the radius to the height of the cone is 24:7.