Solve this radical equation word problem. 16) The area of a given rectangle is \( 120 \mathrm{~cm}^{2} \) If Side \( C \) is 10 cm and Side \( D \) is \( \sqrt{2 x} \) what is the value of \( x ? \) \( \begin{array}{ll}\text { A) } x=12 \mathrm{~cm} & \text { B) } x=120 \mathrm{~cm} \\ \text { C) } x=144 \mathrm{~cm} & \text { D) } x=72 \mathrm{~cm}\end{array} \)

To solve for \( x \) in the radical equation, we start with the formula for the area of a rectangle, which is \( \text{Area} = \text{Length} \times \text{Width} \). Here, the area is \( 120 \mathrm{~cm}^{2} \), one side (Side \( C \)) is \( 10 \mathrm{~cm} \), and the other side (Side \( D \)) is \( \sqrt{2x} \). Setting up the equation: \[ 10 \times \sqrt{2x} = 120 \] Now, divide both sides by 10: \[ \sqrt{2x} = 12 \] Next, square both sides: \[ 2x = 144 \] Dividing by 2 gives: \[ x = 72 \] So the value of \( x \) is \( 72 \mathrm{~cm} \), which corresponds to option D. Now, let’s dive into some fun facts about rectangles and problem-solving! Rectangles have been used for thousands of years! The ancient Egyptians utilized them not only for creating fields but also in complex architectural designs, proving that this shape is fundamental to human innovation. Who knew math could be such an ancient craft? 📜✨ When tackling radical equations, one common mistake people make is forgetting to check for extraneous solutions after squaring both sides. Always plug your solution back into the original equation to ensure it holds true! Trust me, your future self will thank you for that double-check! 🔍😄