Questions: What is the maximum area (in terms of cm^2 ) of a trapezoid placed inside a semicircle with a radius of 1 cm?

Transcript text: H.W.: What is the maximum area (in terms of $\mathrm{cm}^{2}$ ) of a trapezoid placed inside a semicircle with a radius of 1 cm ?

Solution

Solution Steps

Step 1: Understand the Problem

We need to find the maximum area of a trapezoid that can be inscribed in a semicircle with a radius of 1 cm. The semicircle is the upper half of a circle with center at the origin and radius 1 cm.

Step 2: Set Up the Problem

The equation of the semicircle is given by:

\[ x^2 + y^2 = 1, \quad y \geq 0 \]

A trapezoid inscribed in this semicircle will have its vertices on the semicircle. Let's denote the vertices of the trapezoid as $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$, and $D(x_4, y_4)$, where $A$ and $D$ are on the diameter of the semicircle, and $B$ and $C$ are on the arc.

Step 3: Express the Area of the Trapezoid

The area $A$ of a trapezoid with bases $b_1$ and $b_2$ and height $h$ is given by:

\[ A = \frac{1}{2} (b_1 + b_2) \cdot h \]

For the trapezoid inscribed in the semicircle, the bases are the segments $AD$ and $BC$, and the height is the perpendicular distance from $BC$ to the diameter.

Step 4: Maximize the Area

To maximize the area, we need to express the area in terms of the coordinates of the points and use calculus to find the maximum. However, due to symmetry and geometric properties, the maximum area is achieved when the trapezoid is isosceles with $B$ and $C$ at the highest point of the semicircle.

Step 5: Calculate the Maximum Area

For an isosceles trapezoid inscribed in the semicircle, the height $h$ is the radius of the semicircle, which is 1 cm. The bases $b_1$ and $b_2$ are equal, and the maximum occurs when the trapezoid becomes a rectangle with $b_1 = b_2 = 1$.

Thus, the maximum area is:

\[ A = \frac{1}{2} (1 + 1) \cdot 1 = 1 \, \text{cm}^2 \]