The vectors $\mathbf{A}$ and $\mathbf{B}$ each have a magnitude of 5. $\mathbf{A}$ is horizontal and $B$ is at an angle of $45^{\circ}$ to the horizontal. What is the magnitude of $C$, where $C=A+B ?\left(\right.$ Both the sine and the cosine of $45^{\circ}$ are 0.71.) A B

Step 1 ：Given vectors \(\mathbf{A}\) and \(\mathbf{B}\) each have a magnitude of 5. \(\mathbf{A}\) is horizontal and \(\mathbf{B}\) is at an angle of \(45^\circ\) to the horizontal. We are asked to find the magnitude of \(\mathbf{C}\), where \(\mathbf{C}=\mathbf{A}+\mathbf{B}\).

Step 2 ：Since \(\mathbf{A}\) is horizontal and \(\mathbf{B}\) is at an angle of \(45^\circ\) to the horizontal, the components of \(\mathbf{B}\) along the horizontal and vertical directions are both \(5\cos(45) = 5\times0.71\).

Step 3 ：The horizontal component of \(\mathbf{C}\) is the sum of the horizontal components of vectors \(\mathbf{A}\) and \(\mathbf{B}\), which is \(5 + 5\times0.71\).

Step 4 ：The vertical component of \(\mathbf{C}\) is the same as the vertical component of \(\mathbf{B}\), which is \(5\times0.71\).

Step 5 ：The magnitude of \(\mathbf{C}\) is the square root of the sum of the squares of its horizontal and vertical components, which is \(\sqrt{(5 + 5\times0.71)^2 + (5\times0.71)^2}\).

Step 6 ：Calculating the above expression, we find that the magnitude of \(\mathbf{C}\) is approximately 9.24.

Step 7 ：Final Answer: The magnitude of vector \(\mathbf{C}\) is \(\boxed{9.24}\).