To find the height of the ramp \( h \), we can start by using the given equation:
\[ 265 = \sqrt{h^2 + (h + 241)^2} \]
We will first square both sides to eliminate the square root:
\[ 265^2 = h^2 + (h + 241)^2 \]
Calculating \( 265^2 \):
\[ 265^2 = 70225 \]
Now substituting into the equation:
\[ 70225 = h^2 + (h + 241)^2 \]
Next, we expand \((h + 241)^2\):
\[ (h + 241)^2 = h^2 + 2 \cdot h \cdot 241 + 241^2 \]
Calculating \( 241^2 \):
\[ 241^2 = 58081 \]
Now substituting back into the equation gives us:
\[ 70225 = h^2 + h^2 + 2 \cdot 241 \cdot h + 58081 \]
Combining terms:
\[ 70225 = 2h^2 + 482h + 58081 \]
Subtract \( 58081 \) from both sides:
\[ 70225 - 58081 = 2h^2 + 482h \]
Calculating \( 70225 - 58081 \):
\[ 12144 = 2h^2 + 482h \]
Rearranging gives us:
\[ 2h^2 + 482h - 12144 = 0 \]
To simplify the equation, we can divide everything by 2:
\[ h^2 + 241h - 6072 = 0 \]
Now we can use the quadratic formula:
\[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1 \), \( b = 241 \), and \( c = -6072 \).
Calculating \( b^2 - 4ac \):
\[ b^2 - 4ac = 241^2 - 4(1)(-6072) = 58081 + 24288 = 82369 \]
Now we take the square root:
\[ \sqrt{82369} = 287 \]
Now substituting back into the quadratic formula:
\[ h = \frac{-241 \pm 287}{2} \]
Calculating the two possible values:
\[ h = \frac{-241 + 287}{2} = \frac{46}{2} = 23 \]
\[ h = \frac{-241 - 287}{2} = \frac{-528}{2} = -264 ; (\text{not a valid height}) \]
Thus the height of the ramp \( h \) is:
\[ \boxed{23 \text{ inches}} \]