1. **State the problem:** We need to find the perimeter of a rectangle with height $\frac{5}{8}$ ft and width $1 \frac{1}{3}$ ft. 2. **Recall the formula for the perimeter of a rectangle:** $$P = 2(\text{length} + \text{width})$$ 3. **Convert mixed number to improper fraction:** $$1 \frac{1}{3} = \frac{4}{3}$$ 4. **Add the length and width:** $$\frac{5}{8} + \frac{4}{3} = \frac{5 \times 3}{8 \times 3} + \frac{4 \times 8}{3 \times 8} = \frac{15}{24} + \frac{32}{24} = \frac{47}{24}$$ 5. **Multiply by 2 to find the perimeter:** $$P = 2 \times \frac{47}{24} = \frac{2 \times 47}{24} = \frac{94}{24}$$ 6. **Simplify the fraction:** $$\frac{94}{24} = \frac{\cancel{2}47}{\cancel{2}12} = \frac{47}{12}$$ 7. **Convert to mixed number:** $$\frac{47}{12} = 3 \frac{11}{12}$$ 8. **Include the unit:** The perimeter is $3 \frac{11}{12}$ ft. **Final answer:** $$\boxed{3 \frac{11}{12} \text{ ft}}$$