1. **Problem Statement:** Find the value of $x$ in a right triangle where one angle is $45^\circ$, the base is 15, and the hypotenuse is $x$. 2. **Formula and Rules:** In a right triangle, the cosine of an angle is the ratio of the adjacent side (base) to the hypotenuse: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ For $\theta = 45^\circ$, we use: $$\cos(45^\circ) = \frac{15}{x}$$ 3. **Calculate $x$:** $$x = \frac{15}{\cos(45^\circ)}$$ Since $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, substitute: $$x = \frac{15}{\frac{\sqrt{2}}{2}}$$ 4. **Simplify the expression:** $$x = 15 \times \frac{2}{\sqrt{2}}$$ Use rationalization: $$x = 15 \times \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = 15 \times \frac{2\sqrt{2}}{2}$$ Cancel the 2's: $$x = 15 \times \cancel{\frac{2\sqrt{2}}{\cancel{2}}} = 15 \sqrt{2}$$ 5. **Evaluate the numerical value:** $$x \approx 15 \times 1.414 = 21.21$$ 6. **Answer:** The value of $x$ is approximately 21.2, which corresponds to option D.