A rectangle is inscribed in a circle of radius 2 . See the...

A rectangle is inscribed in a circle of radius $2 .$ See the figure. Let $P=(x, y)$ be the point in quadrant I that is a vertex of the rectangle and is on the circle. (a) Express the area $A$ of the rectangle as a function of $x$ (b) Express the perimeter $p$ of the rectangle as a function of $x$. (c) Graph $A=A(x)$. For what value of $x$ is $A$ largest? (d) Graph $p=p(x) .$ For what value of $x$ is $p$ largest?

A rectangle is inscribed in a circle of radius $2 .$ See the figure. Let $P=(x, y)$ be the point in quadrant I that is a vertex of the rectangle and is on the circle.
(a) Express the area $A$ of the rectangle as a function of $x$
(b) Express the perimeter $p$ of the rectangle as a function of $x$.
(c) Graph $A=A(x)$. For what value of $x$ is $A$ largest?
(d) Graph $p=p(x) .$ For what value of $x$ is $p$ largest?

Key Concepts

Inscribed Figures

This concept involves placing one geometric shape inside another so that all vertices of the inner shape touch the boundary of the outer shape. In this context, a rectangle is inscribed in a circle, which means each vertex of the rectangle lies on the circle. This configuration creates a relationship between the dimensions of the rectangle and the radius of the circle.

Circle Equation

The circle equation, x² + y² = r², establishes a fixed relationship between the x and y coordinates for any point on the circle. This fundamental equation allows one to express one coordinate in terms of the other, which is crucial when relating the positions of the rectangle’s vertices to its dimensions.

Area Function for Rectangles

The area function represents the area of the rectangle as a function of one variable. By using relationships derived from the inscribed nature of the rectangle (often with one vertex in the first quadrant), one can express the rectangle's length and width in terms of a single variable, eventually forming an algebraic function that models the area.

Perimeter Function for Rectangles

Similar to the area function, the perimeter function expresses the total boundary length of the rectangle as a function of one variable. It is derived by relating the rectangle’s side lengths to the chosen variable through the circle’s geometric relationships.

Optimization

Optimization involves finding the maximum or minimum values of a function. In this problem, calculus methods such as differentiation are used to determine the value of the variable that maximizes the area or the perimeter of the rectangle. This process is an essential part of applying calculus to solve real-world geometric problems.

Transcript

00:01 All right, guys, so part a wants us to find or to express the area as a function of x.

00:07 So we are given, or we already know that if we were finding the area of a rectangle, we would use the formula area as equal to base times the height.

00:18 So since we're finding this area as a function of x, we were going to use a of x.

00:26 Now, our base is usually represented as a distance from our origin to the x.

00:31 Intercept but since this goes through two quadrants we will have to take in consideration both sides of our own both sides of our axis so we will say two times x and the same goes from our high our height would be from our origin to our y intercept so and since this goes through two origins or two quadrants i'm sorry we will say two times y now now, we know from giving information that one point of our rectangle touches the actual circle.

01:17 And on our figure, we see that the circle is labeled as x squared plus y squared equals four.

01:26 So we wanted to figure out what y is.

01:29 All we need to do is use that equation to solve a y.

01:32 So x square plus y square equals four and if we solve for y y square is equal to four minus x squared just getting y by itself so y is equal to the square root or four minus x square i'm just going to go in and substitute this into my equation for y so two x times times 2 times the square root of 4 minus x square.

02:16 And if i simplify this, 2x times 2 gives me 4x times the square root of 4 minus x squared.

02:24 So that is our area as a function of x.

02:31 Now, part b asks us to find the parameter as the function of x.

02:35 So if we were finding the parameter of a rectangle, or just a rectangle is still, we will use the equation 2 times the base plus 2 times the height.

02:47 And looking back at part a, we already know what our base and our height is.

02:51 So i'm just going to input the information into part b.

02:55 And since we're finding the parameter as a function of x, i want to be using p of x is equal to 2 times the base, which is 2x...

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