Note: Figures not drawn to scale. The angles shown above are acute and $\sin \left(a^{\circ}\right)=\cos \left(b^{\circ}\right)$. If $a=4 k-22$ and $b=6 k-13$, what is the value of $k$ ? A) 4.5 B) 5.5 C) 12.5 D) 21.5

Step 1 ：Given that $a=4k-22$ and $b=6k-13$, and $a+b=90$

Step 2 ：Substitute the values of $a$ and $b$ into the equation: $(4k-22) + (6k-13) = 90$

Step 3 ：Simplify the equation: $10k - 35 = 90$

Step 4 ：Add 35 to both sides: $10k = 125$

Step 5 ：Divide both sides by 10: $k = \frac{25}{2}$

Step 6 ：\(\boxed{k = 12.5}\)