1. **State the problem:** Solve for $c$ in the equation $$\frac{\sqrt{3}}{2} = \frac{34 \frac{1}{2}}{c}.$$ 2. **Convert the mixed number to an improper fraction:** $$34 \frac{1}{2} = 34 + \frac{1}{2} = \frac{68}{2} + \frac{1}{2} = \frac{69}{2}.$$ 3. **Rewrite the equation with improper fraction:** $$\frac{\sqrt{3}}{2} = \frac{\frac{69}{2}}{c} = \frac{69}{2c}.$$ 4. **Set the two fractions equal:** $$\frac{\sqrt{3}}{2} = \frac{69}{2c}.$$ 5. **Cross multiply to solve for $c$:** $$\sqrt{3} \times 2c = 2 \times 69.$$ 6. **Simplify both sides:** $$2c \sqrt{3} = 138.$$ 7. **Isolate $c$ by dividing both sides by $2 \sqrt{3}$:** $$c = \frac{138}{2 \sqrt{3}}.$$ 8. **Simplify the fraction by canceling common factor 2:** $$c = \frac{\cancel{2} \times 69}{\cancel{2} \sqrt{3}} = \frac{69}{\sqrt{3}}.$$ 9. **Rationalize the denominator:** $$c = \frac{69}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{69 \sqrt{3}}{3}.$$ 10. **Simplify the fraction:** $$c = 23 \sqrt{3}.$$ **Final answer:** $$\boxed{c = 23 \sqrt{3}}.$$