Given:
Radius of circular sheet = 30 cm
Area of sector removed = 10% of the area of the circular sheet

To find:
Ratio of radius and height of the cone made from remaining part of the circular sheet.

Solution:
1. Area of the circular sheet = πr²
Area of the circular sheet = π(30)²
Area of the circular sheet = 900π sq.cm

2. Area of the sector removed = 10% of the area of the circular sheet
Area of the sector removed = (10/100) x 900π
Area of the sector removed = 90π sq.cm

3. Area of the remaining part of the circular sheet = Area of the circular sheet - Area of the sector removed
Area of the remaining part of the circular sheet = 900π - 90π
Area of the remaining part of the circular sheet = 810π sq.cm

4. Let the radius and height of the cone be r and h respectively.

5. Slant height (l) of the cone can be calculated using Pythagoras theorem.
l² = r² + h²

6. The curved surface area (CSA) of the cone is given by πrl

7. The remaining part of the circular sheet is used to make the conical surface. Therefore, CSA of the cone = Area of the remaining part of the circular sheet
πrl = 810π
rl = 810
l = 810/r

8. Substitute the value of l in equation (5)
(810/r)² = r² + h²
656100/r² = r² + h²

9. Area of the sector removed is given by (θ/360) x πr², where θ is the angle of the sector in degrees.
(θ/360) x πr² = 90π
θ = 36°

10. Volume of the remaining part of the circular sheet = Volume of the cone
(1/3)πr²h = (1/3) x (θ/360)πr²l
h = (θ/360)l

11. Substitute the value of l and θ in equation (10)
h = (36/360) x (810/r)
h = 3.6/r

12. Substitute the value of h in equation (8)
656100/r² = r² + (3.6/r)²
656100 = r⁴ + 12.96r²
r⁴ + 12.96r² - 656100 = 0

13. Solve the above quadratic equation using the formula,
r² = (-b ± √(b² - 4ac))/2a
where a = 1, b = 12.96, c = -656100

r² = (-12.96 ± √(12.96² - 4 x 1 x -656100))/2 x 1
r² = (-12.96 ± 256.64)/2
r² = 121.84 or r² = -535.84
Since r cannot be negative, discard the negative value of r.
r