1. **State the problem:** We want to find the dimensions of the rectangle with the largest area that has its base on the x-axis and its other two vertices on the parabola $$y = -9x^2 + 675$$ above the x-axis. 2. **Set up the problem:** Let the rectangle have width $$2x$$ (since it is symmetric about the y-axis) and height $$y$$, where $$y = -9x^2 + 675$$. 3. **Area formula:** The area $$A$$ of the rectangle is given by $$A = \text{width} \times \text{height} = 2x \times y = 2x(-9x^2 + 675)$$. 4. **Express area as a function of $$x$$:** $$ A(x) = 2x(-9x^2 + 675) = -18x^3 + 1350x $$ 5. **Find critical points:** To maximize area, find $$A'(x)$$ and set it to zero. $$ A'(x) = \frac{d}{dx}(-18x^3 + 1350x) = -54x^2 + 1350 $$ Set $$A'(x) = 0$$: $$ -54x^2 + 1350 = 0 $$ $$ -54x^2 = -1350 $$ $$ \cancel{-54}x^2 = \cancel{-1350} \Rightarrow x^2 = \frac{1350}{54} = 25 $$ 6. **Solve for $$x$$:** $$ x = \pm 5 $$ Since $$x$$ represents half the width, take $$x = 5$$. 7. **Find height $$y$$:** $$ y = -9(5)^2 + 675 = -9(25) + 675 = -225 + 675 = 450 $$ 8. **Find full width:** $$ \text{width} = 2x = 2 \times 5 = 10 $$ 9. **Final answer:** The rectangle of largest area has dimensions $$10$$ units by $$450$$ units.