Problem: The area of a triangle is 150cm^2. Its sides are in the ratio 3:4:7. Find its perimeter.

Solution:
To solve this problem, we will use the formula for the area of a triangle:
Area = (base x height)/2
We will also use the fact that the sides of the triangle are in the ratio 3:4:7. Let us call the sides of the triangle 3x, 4x, and 7x.

Step 1: Calculate the height of the triangle.
We know that the area of the triangle is 150cm^2. Therefore:
150 = (base x height)/2
Multiplying both sides by 2 gives:
300 = base x height
We can also express the base in terms of the sides of the triangle:
base = 3x + 4x + 7x = 14x
Substituting this into the equation above gives:
300 = 14x x height
Solving for height gives:
height = 300/14x
height = 21.43/x

Step 2: Calculate the perimeter of the triangle.
We can use the Pythagorean theorem to find the length of the base of the triangle:
(3x)^2 + (4x)^2 = (7x)^2
Simplifying this equation gives:
9x^2 + 16x^2 = 49x^2
25x^2 = 49x^2
x^2 = 25/24
x = 5/sqrt(24)
x = 5/(2sqrt(6))

Now we can find the length of each side of the triangle:
3x = 15/(2sqrt(6))
4x = 20/(2sqrt(6))
7x = 35/(2sqrt(6))

The perimeter of the triangle is the sum of the lengths of the sides:
Perimeter = 3x + 4x + 7x
Perimeter = 26/(2sqrt(6))
Perimeter = 13/sqrt(6)

Therefore, the perimeter of the triangle is 13/sqrt(6) cm.

Conclusion: The perimeter of the triangle is 13/sqrt(6) cm.