Find the radii of the bases of the truncated cone if its lateral surface is 182п cm square, the generatrix is 13 cm, and the height is 5 cm.

The lateral surface area is determined by the formula Sbok = πL (R + r), where R and r are the radii of the larger and smaller base, respectively, and L is the generator. Substituting the data known by the condition, we get:
π * 13 * (R + r) = 182 * π;
R + r = 14.
Consider a right-angled triangle formed by the height of the cone, the generatrix of the cone and the projection of the generatrix onto the lower base. This projection is equal to the difference between the radii R – r. The sum of the squares of the legs is equal to the square of the hypotenuse:
h ^ 2 + (R-r) ^ 2 = L ^ 2;
(R-r) ^ 2 = L ^ 2-h ^ 2 = 13 ^ 2-5 ^ 2 = 169-25 = 144;
R-r = √144 = 12.
We have a system of equations:
1) R + r = 14;
2) R-r = 12.
Solving this system by the method of algebraic addition, we get:
2R = 26;
R = 26/2 = 13 cm.
Substituting the obtained value of R into any of the equations of the system, we find r = 1 cm.