1. **State the problem:** We are given the derivative of the plant height function as $$P'(x) = 1 \cdot 1 + 2 \cdot 73 x - 0 \cdot 078 x^2$$ and need to find the range of values of $x$ for which $$P'(x) > 24$$. 2. **Simplify the derivative:** First, simplify the expression for $P'(x)$. $$P'(x) = 1 + 146x - 0.078x^2$$ 3. **Set up the inequality:** We want to solve $$1 + 146x - 0.078x^2 > 24$$ 4. **Rearrange the inequality:** Subtract 24 from both sides: $$1 + 146x - 0.078x^2 - 24 > 0$$ $$-0.078x^2 + 146x - 23 > 0$$ 5. **Multiply by -1 to make the quadratic coefficient positive:** $$0.078x^2 - 146x + 23 < 0$$ Note the inequality sign flips when multiplying by -1. 6. **Solve the quadratic inequality:** Find roots of $$0.078x^2 - 146x + 23 = 0$$ Using the quadratic formula: $$x = \frac{146 \pm \sqrt{146^2 - 4 \cdot 0.078 \cdot 23}}{2 \cdot 0.078}$$ Calculate discriminant: $$146^2 = 21316$$ $$4 \cdot 0.078 \cdot 23 = 7.176$$ $$\sqrt{21316 - 7.176} = \sqrt{21308.824} \approx 146.0$$ 7. **Calculate roots:** $$x_1 = \frac{146 - 146}{0.156} = 0$$ $$x_2 = \frac{146 + 146}{0.156} = \frac{292}{0.156} \approx 1871.79$$ 8. **Determine the solution interval:** Since the quadratic opens upwards (positive coefficient), the inequality $$0.078x^2 - 146x + 23 < 0$$ holds between the roots: $$0 < x < 1872$$ (rounded to nearest whole number) 9. **Final answer:** The range of $x$ values for which $$P'(x) > 24$$ is $$\boxed{1 \leq x \leq 1871}$$ (We use 1 as the nearest whole number greater than 0.)