1. **State the problem:** We are given a rectangle with height 5 cm and width $r - 4$ cm. The area of the rectangle is 10 cm$^2$. We need to find the value of $r$. 2. **Formula for area of a rectangle:** $$\text{Area} = \text{height} \times \text{width}$$ 3. **Apply the formula:** $$10 = 5 \times (r - 4)$$ 4. **Simplify the equation:** $$10 = 5r - 20$$ 5. **Add 20 to both sides:** $$10 + 20 = 5r - 20 + 20$$ $$30 = 5r$$ 6. **Divide both sides by 5:** $$\frac{30}{\cancel{5}} = \frac{5r}{\cancel{5}}$$ $$6 = r$$ 7. **Final answer:** The value of $r$ is $6$ cm.