1. **State the problem:** We need to find the value of $\tan E$ in the right triangle $EDC$ where $ED=24$, $DC=32$, and hypotenuse $EC=40$. The right angle is at $D$. 2. **Recall the definition of tangent:** In a right triangle, $\tan$ of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. 3. **Identify sides relative to angle $E$:** - Opposite side to angle $E$ is $DC=32$. - Adjacent side to angle $E$ is $ED=24$. 4. **Write the formula:** $$\tan E = \frac{\text{opposite}}{\text{adjacent}} = \frac{DC}{ED} = \frac{32}{24}$$ 5. **Simplify the fraction:** $$\tan E = \frac{32}{24} = \frac{\cancel{32}}{\cancel{24}} = \frac{4}{3}$$ 6. **Calculate the decimal value:** $$\tan E = \frac{4}{3} \approx 1.3333$$ 7. **Round to the nearest hundredth:** $$\tan E \approx 1.33$$ **Final answer:** $\boxed{1.33}$