1. **State the problem:** We need to find the area of a trapezoid with height $4$ cm, shorter base $1 \frac{1}{2}$ cm, and longer base $3 \frac{1}{2}$ cm. 2. **Formula for the area of a trapezoid:** $$\text{Area} = \frac{(\text{base}_1 + \text{base}_2)}{2} \times \text{height}$$ This formula calculates the average length of the two bases multiplied by the height. 3. **Convert mixed numbers to improper fractions:** $$1 \frac{1}{2} = \frac{3}{2}, \quad 3 \frac{1}{2} = \frac{7}{2}$$ 4. **Substitute values into the formula:** $$\text{Area} = \frac{\left(\frac{3}{2} + \frac{7}{2}\right)}{2} \times 4$$ 5. **Add the bases:** $$\frac{3}{2} + \frac{7}{2} = \frac{10}{2} = 5$$ 6. **Simplify the expression:** $$\text{Area} = \frac{5}{2} \times 4$$ 7. **Multiply and simplify by canceling common factors:** $$\text{Area} = \cancel{\frac{5}{2}} \times \cancel{4} = 5 \times 2 = 10$$ 8. **Final answer:** The area of the trapezoid is $10$ square centimeters.