Problem 30 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A rectangle is inscribed in a semicircle of radius \(R\), where \(R\) is constant. (a) Express the area of the rectangle as a function of the height, \(h\), of the rectang \(A=f(h)\) (b) Express the perimeter of the rectangle as a function of the height, \(h\), of rectangle, \(P=g(h)\).

Step-by-Step Solution

Answer

The area of the rectangle as a function of height is given by \(f(h) = 4h^2\) and the perimeter of the rectangle as a function of height is given by \(g(h) = 10h\).

Step 1: Derive the function \(f(h)\)

Express the area of the rectangle \(A\) as a function of the height \(h\). The area of a rectangle is given by the product of its width and its height. In this case, the width of the rectangle, being the diameter of the semicircle, is 2R. Thus, we can express the area \(A\) as \(A = 2Rh\). However, \(R\) is relative to \(h\) because the rectangle is inscribed in the semicircle: \(R = 2h\). Substitute \(R = 2h\) into the area equation to get the function in terms of \(h\), which becomes \(f(h) = 4h^2\).

Step 2: Derive the function \(g(h)\)

Express the perimeter of the rectangle \(P\) as a function of the height \(h\). The perimeter of a rectangle is given by twice the sum of its width and height. In this case, the perimeter \(P\) can be expressed as \(P = 2*(2R + h)\). We know that \(R = 2h\), so we substitute this in to get \(P = 2*(4h + h)\), which simplifies to \(g(h) = 10h\).

Key Concepts

Understanding Rectangles

A rectangle is a four-sided polygon, often referred to as a quadrilateral. It has several important properties which make it distinct:

• All its angles are right angles, meaning each angle measures 90 degrees.
• The opposite sides are equal in length and parallel.
• The area of a rectangle is calculated by multiplying its width by its height: Area = Width x Height.

A rectangle becomes part of intriguing geometric problems, especially when inscribed in other shapes, such as circles or semicircles. When inscribed in a semicircle, the rectangle's width can sometimes be described using the diameter of the semicircle.In our example, the rectangle is inscribed in a semicircle with radius \(R\). By determining how the geometry confines its dimensions, you can derive expressions for both its area and perimeter with respect to one of its dimensions – the height \(h\). In this setup, the width and height are interrelated by the semicircle's geometry, leading to specific mathematical expressions.

Exploring the Semicircle

A semicircle is essentially half of a circle. It contains some unique properties that align with the circle:

• The semicircle's diameter is equal to the full diameter of the circle it was split from, which is twice the radius \(R\).
• The semicircle has a 180-degree arc, half the circle's 360-degree complete rotation.
• The area of the semicircle can be calculated by taking half the area of the circle.

In problems involving inscribed shapes, the semicircle often serves as a boundary that confines or determines the dimensions of the inscribed shape. For example, if a rectangle is inscribed in a semicircle, its width cannot exceed the semicircle's diameter, directly linking the dimensions of the rectangle to the radius of the semicircle.In our exercise, the rectangle's height is variable and used to express both the area and perimeter, while the width depends on the semicircle's diameter, impacted by this radius \(R\). Consequently, the semicircle impacts the algebraic expressions of the inscribed rectangle.

Defining the Radius

The radius \(R\) is a foundational element in circular and semicircular geometry. It is the distance from the center of the circle or semicircle to any point on its perimeter. Here are some notable facts:

• The radius determines the circle's overall scale, affecting its diameter, area, and circumference.
• In a semicircle, the radius helps specify the span since it defines the semicircle's reach.
• The length of the radius factors into many geometric equations, such as circumference \(C = 2\pi R\) and area \(A = \pi R^2\) for circles, adapted in the semicircle scenario.

When dealing with inscribed rectangles, the radius \(R\) becomes a pivotal measurement. In our problem, it regulates the rectangle's maximum possible width, since width equals the semicircle's diameter or \(2R\). This demands that any expression for the rectangle's area or perimeter critically uses \(R\), showcasing its essential role in calculating geometric properties. Moreover, understanding the radius and its applications across various shapes facilitates a deeper comprehension of geometry problems and solutions.