Let's start by using the formula for the surface area of a rectangular prism, which is:
SA = 2lw + 2lh + 2wh
where SA is the surface area, l is the length, w is the width, and h is the height of the rectangular prism.
We know that the surface area of the rectangular prism is 174 square inches, so we can write:
174 = 2lw + 2lh + 2wh
We also know that one side length of the rectangular base is 3 times the other. Let's call the shorter side length "x" and the longer side length "3x". Then, we can write:
l = 3x
w = x
Now we can substitute these values into the surface area formula:
174 = 2(3x)(x) + 2(3x)(h) + 2(x)(h)
Simplifying this expression, we get:
174 = 6x^2 + 6xh
Now we can solve for h:
h = (174 - 6x^2) / (6x)
We can simplify this expression by factoring out a 6 from the numerator:
h = (29 - x^2) / x
So the height of the rectangular prism is (29 - x^2) / x.