Solution:

Let's solve this step by step.

Step 1: Given that tan(A) = 1/2 and tan(B) = 1/3, we need to find the value of tan(A+B).

Step 2: To find tan(A+B), we can use the formula:

tan(A+B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))

Step 3: Substituting the given values, we have:

tan(A+B) = (1/2 + 1/3)/(1 - (1/2)(1/3))

Step 4: Simplifying the expression further:

tan(A+B) = (3/6 + 2/6)/(1 - 1/6)

tan(A+B) = 5/6 / 5/6

tan(A+B) = 1

Step 5: We have found that tan(A+B) = 1. Now, we need to find the angle A+B.

Step 6: We know that the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.

Step 7: Since A and B are acute angles, we can consider right triangles where A and B are the angles.

Step 8: Let's consider a right triangle ABC, where angle A is the acute angle and the side opposite to angle A is x, and the side adjacent to angle A is 2x.

Step 9: Using the Pythagorean theorem, we can find the hypotenuse of the triangle:

(2x)^2 + x^2 = h^2

4x^2 + x^2 = h^2

5x^2 = h^2

h = x√5

Step 10: Now, let's consider another right triangle DEF, where angle B is the acute angle and the side opposite to angle B is y, and the side adjacent to angle B is 3y.

Step 11: Using the Pythagorean theorem, we can find the hypotenuse of the triangle:

(3y)^2 + y^2 = h^2

9y^2 + y^2 = h^2

10y^2 = h^2

h = y√10

Step 12: Since tan(A) = 1/2, we can write:

x/(2x) = 1/2

x = 2

Step 13: Since tan(B) = 1/3, we can write:

y/(3y) = 1/3

y = 3

Step 14: Now, we can find the value of x√5 and y√10:

x√5 = 2√5

y√10 = 3√10

Step 15: We know that h = x√5 = y√10, so: