https://file.aiquickdraw.com/mathbot/file/172865826629412xwjkwq.jpg

Let's break down the problem from the image you provided:

The problem asks about a rectangle with the largest possible area that can be inscribed in a semicircle with a radius of 2 cm. The task is to find the ratio of the lengths of the largest side to the smallest side of this rectangle.

### Step 1: Geometry Setup
Let the radius of the semicircle be \( r = 2 \, \text{cm} \). The largest rectangle that can be inscribed in a semicircle is known to be a **square** when maximizing area.

We know from geometric properties that the ratio of the largest side to the smallest side for the maximum-area rectangle inscribed in a semicircle is always \( \sqrt{2}:1 \).

### Step 2: Answer
Thus, the correct ratio of the lengths of the largest to the smallest side of this rectangle is:

\[
\boxed{\sqrt{2}:1}
\]

This matches **Option 1** in the question.

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Would you like a more detailed explanation of why the maximum-area rectangle in a semicircle is a square?

### Related Questions:
1. How can we derive the formula for the area of a rectangle inscribed in a semicircle?
2. What is the equation of the semicircle if its radius is arbitrary?
3. How do we calculate the derivative to maximize the area of the inscribed rectangle?
4. Can this problem be generalized for ellipses instead of circles?
5. What is the relationship between the perimeter and the area of the inscribed rectangle?

**Tip**: Maximization problems in geometry often require derivatives, so keep calculus concepts in mind when dealing with similar optimization tasks!

A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

Find the ratio of the largest to smallest side of the maximum-area rectangle inscribed in a semicircle with a radius of 2 cm. This problem involves geometric optimization and is suitable for students in Grades 8-10.

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