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Step-by-Step Solution
Step 1: Understand the Geometric Setup
We have a parabola given by $y = 12 - x^2$. We want to inscribe a rectangle inside this parabola such that its base lies on the x-axis and its upper corners lie on the parabola. Let the x-coordinates of these points be $-x$ and $x$, which makes the base length $2x$.
Step 2: Express the Height of the Rectangle
The top vertices of the rectangle at $(x, y)$ and $(-x, y)$ both lie on the parabola, so their common $y$-value is $y = 12 - x^2$.
Step 3: Write Down the Area Function
Since the base of the rectangle is $2x$ and its height is $y = 12 - x^2$, the area of the rectangle $A$ is:
$A(x) = \text{(base)} \times \text{(height)} = 2x \times (12 - x^2) = 24x - 2x^3.$
Step 4: Differentiate to Find the Critical Points
To find the value of $x$ that maximizes the area, we differentiate $A(x)$ with respect to $x$ and set it to zero:
$\frac{dA}{dx} = 24 - 6x^2.$
Set $\frac{dA}{dx} = 0$ to find critical points:
$24 - 6x^2 = 0.$
$6x^2 = 24.$
$x^2 = 4.$
$x = 2$ (taking the positive value since $x$ represents a length from the origin).
Step 5: Compute the Maximum Area
Substitute $x = 2$ into the expression for $A(x)$ to find the corresponding area:
$A(2) = 2(2)\,(12 - 2^2) = 4 \times (12 - 4) = 4 \times 8 = 32.$
Thus, the maximum area of the rectangle is $32\text{ sq. units}$.
Step 6: Verify the Result Visually (Provided Image)
Final Answer
The maximum area of the rectangle is 32 square units.
Step-by-Step Solution
Step 1: Understand the Geometric Setup
We have a parabola given by $y = 12 - x^2$. We want to inscribe a rectangle inside this parabola such that its base lies on the x-axis and its upper corners lie on the parabola. Let the x-coordinates of these points be $-x$ and $x$, which makes the base length $2x$.
Step 2: Express the Height of the Rectangle
The top vertices of the rectangle at $(x, y)$ and $(-x, y)$ both lie on the parabola, so their common $y$-value is $y = 12 - x^2$.
Step 3: Write Down the Area Function
Since the base of the rectangle is $2x$ and its height is $y = 12 - x^2$, the area of the rectangle $A$ is:
$A(x) = \text{(base)} \times \text{(height)} = 2x \times (12 - x^2) = 24x - 2x^3.$
Step 4: Differentiate to Find the Critical Points
To find the value of $x$ that maximizes the area, we differentiate $A(x)$ with respect to $x$ and set it to zero:
$\frac{dA}{dx} = 24 - 6x^2.$
Set $\frac{dA}{dx} = 0$ to find critical points:
$24 - 6x^2 = 0.$
$6x^2 = 24.$
$x^2 = 4.$
$x = 2$ (taking the positive value since $x$ represents a length from the origin).
Step 5: Compute the Maximum Area
Substitute $x = 2$ into the expression for $A(x)$ to find the corresponding area:
$A(2) = 2(2)\,(12 - 2^2) = 4 \times (12 - 4) = 4 \times 8 = 32.$
Thus, the maximum area of the rectangle is $32\text{ sq. units}$.
Step 6: Verify the Result Visually (Provided Image)
Final Answer
The maximum area of the rectangle is 32 square units.
