# The height of the cylinder is 10 cm greater than the radius of the base, and the total

**The height of the cylinder is 10 cm greater than the radius of the base, and the total surface is 144п cm. Determine the radius of the base and the height.**

The total surface area of the cylinder is equal to the sum of the areas of the lateral surface and two bases:

S full = S side + 2 * S main.

The base area of the cylinder is determined by the formula:

Sbn = πR ^ 2, where R is the radius of the base.

The area of the lateral surface of the cylinder is equal to the product of the circumference of the base by the height:

Sside = 2πR * h.

By condition, h = R + 10 and Stot = 144π, we can write the equation:

2πR * h + 2 * πR ^ 2 = 144π;

2πR * (R + 10) + 2 * πR ^ 2 = 144π;

2πR ^ 2 + 2πR ^ 2 + 20πR = 144π;

4πR ^ 2 + 20πR – 144π = 0.

Divide both sides of the equation by 4n, we get:

R ^ 2 + 5R – 36 = 0.

D = 52 – 4 * (- 36) = 25 + 144 = 169 = 132;

R1 = (- 5 – 13) / 2 = – 18/2 = – 9 – does not satisfy the condition of this problem, since the radius cannot take negative values.

R2 = (- 5 + 13) / 2 = 8/2 = 4.

h = R + 10 = 4 + 10 = 14.

The radius of the base of this cylinder is 4 cm, the height of the cylinder is 14 cm.