The accompanying figure shows a circular sector with radius r cm and central angle θ(radian meas

The accompanying figure shows a circular sector with radius...

Key Concepts

Circular Sector

A circular sector is a portion of a circle defined by two radii and the arc between them. Its properties include the radius, the central angle (usually measured in radians), the arc length, and the area. Understanding these features is crucial when analyzing and solving problems related to sectors.

Perimeter of a Circular Sector

The perimeter of a circular sector consists of the lengths of the two radii and the arc. The arc length is computed as the product of the radius and the central angle (when the angle is given in radians). This relationship is key to setting up equations that relate the dimensions of the sector, such as when a fixed perimeter is provided.

Area of a Circular Sector

The area of a circular sector is determined by the formula (1/2)*r^2*?, which reflects the proportion of the circle's area corresponding to the central angle in radians. This concept is fundamental in relating the area to the radius and the central angle, and it supports further exploration such as determining function relationships or optimizing the area.

Function Relationships in Geometric Contexts

Establishing function relationships involves expressing one variable in terms of another using given constraints. In geometric problems, this often means deriving formulas that express the radius as a function of the central angle or vice versa, based on relationships such as the perimeter or area formulas.

Quadratic Functions and Their Characteristics

Quadratic functions are defined by polynomial expressions of degree two, typically taking the form ax^2 + bx + c, and graph as parabolas. Recognizing and understanding quadratic functions, including their vertex and symmetry properties, is important when determining the nature of relationships and whether a derived function exhibits quadratic behavior.

Optimization in Geometry

Optimization problems in geometry involve finding maximum or minimum values of a function under given constraints, such as maximizing the area of a circular sector with a fixed perimeter. This often requires setting up and solving equations derived from geometric properties, and then applying techniques such as completing the square or using calculus to find the optimal values.

Transcript

00:02 We're asked to answer a question about maximum area using optimization.

00:07 So we're told to consider a symmetric cross inscribed in a circle of radius r.

00:12 This is shown in the figure of exercise 52.

00:17 In part a, we're asked to write the area a of the cross as a function of x and find the value of x that maximizes the area.

00:28 To find the area, first let's find the width of the arm with x in zero or well, the width of this arm by pythagorean theorem, this is 2 times the square root of r squared minus x squared.

00:50 Then working from left to right, we have that our area is equal to the horizontal arm of the rectangle, which is 2 times the square root of r squared minus x squared times x minus x minus, square root of r squared minus x squared so that's the area of that horizontal arm then to this we want to add the vertical rectangle well this is two times the square root of r squared minus x squared times two x and finally we add the area of the other horizontal arm which once again is two times the square root of r squared minus x squared times x minus the square root of r squared minus x squared.

02:07 And so we can simplify this as the area as a function of x is equal to four times the square root of r squared minus x squared times 2x minus the square root of r squared minus x squared which we could also write this as four times and this is 2x times and this is 2x times the square root of r squared minus x squared minus r squared minus x squared minus x squared a prime of x squared.

03:25 Now to find the value of x that maximizes the area, we want to find the first derivative, a prime of x, this is equal to 8 times the square root of r squared minus x squared.

03:51 Minus x squared over the square root of r squared minus x squared plus x.

04:09 So it factored out a two.

04:21 And we can also write this as 8 over the square root of r squared minus x squared times r squared minus x squared minus x squared plus x times square root of r squared minus x squared minus x squared.

04:46 Now we want to find the critical values.

04:51 So we want to set this equal to zero.

04:59 So we want to solve the equation.

05:06 X times the square root of r squared minus x squared equals negative r squared plus 2x squared.

05:16 To solve this radical equation, i'm going to square the entire equation.

05:22 So i get x squared times r squared minus x squared.

05:28 And on the right hand side, i get 4x to the fourth minus 4 r squared x squared plus r squared x squared plus r to the fourth.

05:55 And to make this simpler, i'll introduce the new variable z equals x squared over r squared.

06:05 If i divide through by r to the fourth, i get, this is z times 1 over r squared minus.

06:19 Actually this is just one minus z equals 4 times z squared minus 4 times z plus 1 and so we obtained the equation 4 z squared plus z squared this is 5 z squared minus 5 z squared minus 5 z squared plus 1 equals 0.

07:04 And if you solve this using quadratic formula, we get that z equals 5 plus or minus the square root of 5 over 10.

07:19 Both of these are valid answers, since they're both positive.

07:42 And so i need you to admit.

07:51 Why am i a mark when i see one? and so we solved for x.

07:57 And so we solved for x.

08:00 Is it open all of you? x squared over r squared equals 5 plus or minus root 5 over 10 so that x is equal to r plus or minus r times the square root of 5 plus or minus root 5 over 10.

08:35 Of course, from our definition, x has to be positive.

08:41 So we x is positive r times the square root of 5 plus or minus root of 5 plus or minus root 5 over 10...

Please give Ace some feedback