What is the solution of $(4x - 16)^{\frac{1}{2}} = 36$?
The problem requires solving the equation \((4x - 16)^{\frac{1}{2}} = 36\) for the variable \(x\). First, note that \((4x - 16)^{\frac{1}{2}}\) is equivalent to the principal square root \(\sqrt{4x - 16}\). For this expression to be defined in real numbers, the radicand must satisfy the condition \(4x - 16 \geq 0\), which implies \(x \geq 4\).
To solve \(\sqrt{4x - 16} = 36\), we eliminate the square root by squaring both sides of the equation:
\[(\sqrt{4x - 16})^2 = 36^2\]
This gives us:
\[4x - 16 = 1296\]
Now, isolate the term containing \(x\) by adding \(16\) to both sides:
\[4x = 1296 + 16\]
\[4x = 1312\]
Divide both sides by \(4\) to find \(x\):
\[x = \frac{1312}{4} = 328\]
Finally, we verify the solution \(x = 328\) by substituting it back into the original equation:
\[\sqrt{4(328) - 16} = \sqrt{1312 - 16} = \sqrt{1296}\]
Since \(36 \times 36 = 1296\), we have \(\sqrt{1296} = 36\). The equation holds true, and \(328 \geq 4\), so the solution is valid. The correct option is \(x = 328\).
This type of equation appears in many real-world contexts, such as calculating distance or scale. For instance, if you are designing a square garden and you know the area is 1296 square feet, the side length \(s\) is \(\sqrt{1296} = 36\) feet. If the side length is defined by a formula like \(\sqrt{4x-16}\), you can use the same algebraic steps to find the value of \(x\) that corresponds to the desired area, allowing you to correctly adjust your layout.