Question

find the dimensions of the rectangle of largest area that has its base on the x - axis and its other 2 vertices above the x - axis and lying on the parabola $y = - 3x^2 + 225$.

Explanation:

Step1: Define variables

Let the right vertex on the parabola be $(x, y)$ where $y = -3x^2 + 225$. The base of the rectangle is $2x$, height is $y$.

Step2: Set area function

Area $A(x) = 2x \cdot y = 2x(-3x^2 + 225) = -6x^3 + 450x$

Step3: Find critical points

Take derivative: $A'(x) = -18x^2 + 450$
Set $A'(x)=0$:
$$-18x^2 + 450 = 0$$
$$18x^2 = 450$$
$$x^2 = 25$$
$$x = 5 \quad (x>0 \text{ as it's right vertex})$$

Step4: Verify maximum

Second derivative: $A''(x) = -36x$. At $x=5$, $A''(5) = -180 < 0$, so it's a maximum.

Step5: Find height and dimensions

Calculate $y$: $y = -3(5)^2 + 225 = -75 + 225 = 150$
Base length: $2x = 10$

Answer:

The rectangle has a base of length $10$ and height of $150$, so the dimensions are $\boldsymbol{10 \times 150}$.