What expression represents the area of the rectangle inscribed in the semicircle?
Understand the Problem
The question is asking for the mathematical expression that defines the area of a rectangle that is inscribed in a semicircle. To solve it, we need to apply concepts of geometry related to rectangles and semicircles to determine which expression accurately represents the area of the rectangle based on its dimensions.
Answer
The area of a rectangle inscribed in a semicircle is given by the expression $A(x) = 2x \cdot \sqrt{r^2 - x^2}$.
Answer for screen readers
The area of a rectangle inscribed in a semicircle is given by the expression:
$$ A(x) = 2x \cdot \sqrt{r^2 - x^2} $$
Steps to Solve
- Identify the shape and dimensions
Let's denote the radius of the semicircle as $r$. The rectangle inscribed in the semicircle will have its width as $2x$, where $x$ is the horizontal distance from the center of the semicircle to one of the rectangle's vertical sides. The height of the rectangle will be $y$.
- Use the equation of the semicircle
In a semicircle with radius $r$ centered at the origin, the equation is given by
$$ y = \sqrt{r^2 - x^2} $$
Here, $y$ is the height of the rectangle determined by the semicircle's equation, while $x$ measures the horizontal distance.
- Express the area of the rectangle
The area $A$ of the rectangle can be expressed as:
$$ A = \text{width} \times \text{height} = (2x)(y) $$
Substituting $y$ from the semicircle equation gives:
$$ A = 2x \cdot \sqrt{r^2 - x^2} $$
- Simplify the area expression
Now we have the area expression of the rectangle:
$$ A(x) = 2x \cdot \sqrt{r^2 - x^2} $$
This is the mathematical expression that defines the area of the rectangle inscribed in the semicircle in terms of $x$.
The area of a rectangle inscribed in a semicircle is given by the expression:
$$ A(x) = 2x \cdot \sqrt{r^2 - x^2} $$
More Information
This expression shows that the area of the rectangle depends on the distance $x$ from the center to the edge of the rectangle. By maximizing this area function, we can find the optimal dimensions of the rectangle for the largest area.
Tips
- Not correctly using the semicircle equation. Ensure to remember that it is $y = \sqrt{r^2 - x^2}$ and not $y = r - x$.
- Forgetting to multiply by 2 for the width when using $x$ as half the width.
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