1. **State the problem:** We need to find the tangent of angle $Q$ in right triangle $PQR$ where $\angle R$ is the right angle. 2. **Recall the definition of tangent:** $$\tan(Q) = \frac{\text{opposite side to } Q}{\text{adjacent side to } Q}$$ 3. **Identify the sides relative to $Q$:** - Opposite side to $Q$ is $PR = 2\sqrt{10}$ - Adjacent side to $Q$ is $RQ = \sqrt{30}$ 4. **Calculate $\tan(Q)$:** $$\tan(Q) = \frac{2\sqrt{10}}{\sqrt{30}}$$ 5. **Simplify the fraction:** $$\tan(Q) = \frac{2\sqrt{10}}{\sqrt{30}} = \frac{2\sqrt{10}}{\sqrt{30}} \times \frac{\sqrt{30}}{\sqrt{30}} = \frac{2\sqrt{10} \times \sqrt{30}}{30}$$ 6. **Multiply under the square root:** $$\sqrt{10} \times \sqrt{30} = \sqrt{10 \times 30} = \sqrt{300}$$ 7. **Simplify $\sqrt{300}$:** $$\sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3}$$ 8. **Substitute back:** $$\tan(Q) = \frac{2 \times 10 \sqrt{3}}{30} = \frac{20\sqrt{3}}{30}$$ 9. **Simplify the fraction by canceling common factors:** $$\tan(Q) = \frac{\cancel{20}\sqrt{3}}{\cancel{30}} = \frac{2\sqrt{3}}{3}$$ 10. **Calculate decimal approximation:** $$\tan(Q) \approx \frac{2 \times 1.732}{3} = \frac{3.464}{3} \approx 1.15$$ **Final answer:** $$\tan(Q) = \frac{2\sqrt{3}}{3} \approx 1.15$$