Problem 58 - Calculus with Concepts in Calculus

Question

A rectangle is inscribed in a semicircle of radius \(r\) with one side lying on the diameter of the semicircle. Find the maximum possible area of the rectangle.

Step-by-Step Solution

Answer

The maximum possible area of the rectangle is \( r^2 \).

Step 1: Understand the Problem

We need a rectangle inscribed in a semicircle with radius \( r \), where one side of the rectangle rests on the diameter of the semicircle. Our task is to find the dimensions that give this rectangle the maximum area.

Step 2: Set Up the Geometry

Let the rectangle have its base on the diameter and its top corners touching the semicircle. Let the width of the rectangle be \( 2x \) and the height \( y \). The relationship comes from the Pythagorean theorem: \( x^2 + y^2 = r^2 \).

Step 3: Express Area in Terms of Variables

The area \( A \) of the rectangle is \( A = 2x \times y \). We need to express \( y \) in terms of \( x \) using the previous step: \( y = \sqrt{r^2 - x^2} \). Thus, the area is \( A = 2x\sqrt{r^2 - x^2} \).

Step 4: Differentiate the Area Function

Differentiate the area function \( A(x) = 2x\sqrt{r^2 - x^2} \) with respect to \( x \) for finding maximum. To differentiate, use the product and chain rules.

Step 5: Solve for Critical Points

Set the derivative \( \frac{dA}{dx} = 0 \) to find critical points. The equation simplifies to \( \sqrt{r^2 - x^2} = \frac{x^2}{r^2 - x^2} \). Solving gives \( x = \frac{r}{\sqrt{2}} \).

Step 6: Verify Maximum Using the Second Derivative Test

Differentiate again to find \( \frac{d^2A}{dx^2} \) and evaluate it at \( x = \frac{r}{\sqrt{2}} \). Since this value is negative, it confirms a local maximum.

Step 7: Substitute Back to Find Maximum Area

Substitute \( x = \frac{r}{\sqrt{2}} \) back into the area function \( A \). \( y \) becomes \( \frac{r}{\sqrt{2}} \) as well. Therefore, the maximal area \( A = 2 \left( \frac{r}{\sqrt{2}} \right) \left( \frac{r}{\sqrt{2}} \right) = r^2 \).

Key Concepts

Inscribed Shapes

When discussing inscribed shapes, we're talking about one shape fitting snugly inside another. In this case, a rectangle is inscribed in a semicircle. This means the rectangle's base lies on the semicircle's diameter, and its upper corners touch the arc of the semicircle itself.
Visualizing this can help in understanding the constraints of the problem. Imagine a semicircle with its flat side facing up, and then picture a rectangle resting inside so that its width covers the straight line across the diameter. The height of this rectangle will rise to meet the curve of the semicircle.
Inscribed shapes appear frequently in geometry problems where one must maximize or minimize an area or perimeter under given conditions. In this exercise, we're particularly interested in maximizing the area of the rectangle while it remains cornfirmed within the constraints of the semicircle.

Area Maximization

Area maximization refers to the process of finding the dimensions that provide the largest possible area for a shape under given constraints. In our scenario, we want to maximize the area of the inscribed rectangle.
To achieve this, we begin by looking at a rectangle's area, given as Area = width × height. For a rectangle inscribed in a semicircle, let the width be twice some value, 2x, and the height as y. The formula for area then becomes:

• Area = 2x × y

The unique challenge posed by the semicircle wraps around using the side and endpoints to provide limits. But this also means the relations of x and y are bound by the semicircle’s boundary.
It requires setting the problem up as a function of a variable, and applying calculus techniques such as differentiation to find maximum values. We differentiate the area function with respect to x and find critical points where the derivative equals zero. From these points, we evaluate which configuration gives the largest area.
This involves checking that the critical point indeed gives a maximum by, for example, applying a second derivative test.

Pythagorean Theorem

The Pythagorean Theorem is essential in optimizing problems involving shapes like our inscribed rectangle in a semicircle. This theorem links the sides of right-angled triangles, stating that for a triangle with perpendicular sides x, y, and hypotenuse r, the relation is:

• \[ x^2 + y^2 = r^2 \]

In the context of a rectangle inscribed in a semicircle, we can use the Pythagorean theorem to describe the relationship between base, height, and radius of the semicircle. Here, x is half of the base of the rectangle, y is the height, and r is the semicircle's radius.
By rearranging this geometric relation, we find y, expressed as a function of x:

• \[ y = \sqrt{r^2 - x^2} \]

This given relationship between x and y is paramount in expressing the area function of the rectangle and finding its maximum. Understanding the Pythagorean Theorem helps us to explore how geometric constraints help structure and simplify complex optimization problems.