Math Is Fun Forum

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 687

Express Total Area A

A wire 10 meters long is to be cut into two pieces. One piece will be shaped like a square, and the other piece will be shaped as a circle.

Wire length = 10 meters

Piece 1 = 4x

Piece 2 = 10 - 4x

Express the total area A enclosed by the pieces if wire as a function of the length x of a side of the square.

Let me see.

We want A(x).

A(x) = (side)^2/(total number of sidea) + (side)^2/(total number of sides)

A(x) = (x)^2/4 + (10 - 4x)^2/4

A(x) = (x^2/4) + (100 - 80x + 16x^2)/4

A(x) = (17x^2 - 80x + 100)/4

Is this right?

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The greatest truth about the Rapture is not its timing but it's reality.

Dr. David Jeremiah

Bob

Administrator

Registered: 2010-06-20

Posts: 10,279

Re: Express Total Area A

Wire length = 10 meters

Piece 1 = 4x

Piece 2 = 10 - 4x

Ok so far.

Area of a square = side squared = x^2

area of circle = pi r^2

Here we don't know the radius but we do know the circumference 10 - 4x

C = 2pi.r  => r = C/(2pi) = (10-4x)/(2pi)

So area = pi [(10-4x)/(2pi)]^2

Add the areas.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 687

Re: Express Total Area A

Wire length = 10 meters

Piece 1 = 4x

Piece 2 = 10 - 4x

Ok so far.

Area of a square = side squared = x^2

area of circle = pi r^2

Here we don't know the radius but we do know the circumference 10 - 4x

C = 2pi.r  => r = C/(2pi) = (10-4x)/(2pi)

So area = pi [(10-4x)/(2pi)]^2

Add the areas.

Bob

A(x) = x^2 + pi•[(10 - 4x)/(2•pi)]^2

A(x) = x^2 + pi(100 - 80x + 16x^2)/(pi^2)

A(x) = x^2 + (25 - 20x + 4x^2)/(pi)

You say?

Also, what is the domain of A?

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The greatest truth about the Rapture is not its timing but it's reality.

Dr. David Jeremiah

Bob

Administrator

Registered: 2010-06-20

Posts: 10,279

Re: Express Total Area A

That value of A is correct.

I then put that equation into the graph plotter to see what it looks like. Basically its a quadratic so it's a U-shaped parabola.  The minimum point is needed if you want the range for A. For the domain of x, ask what are the minimum and maximum values x can have in this problem.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 687

Re: Express Total Area A

That value of A is correct.

I then put that equation into the graph plotter to see what it looks like. Basically its a quadratic so it's a U-shaped parabola.  The minimum point is needed if you want the range for A. For the domain of x, ask what are the minimum and maximum values x can have in this problem.

Bob

Ok. I will also graph this model using Desmos. I want and need to be able to find the domain and range from the information given in a graph. Without this skill, I better not even think about moving forward in my self-study of mathematics. I really hate wasting time.

Also, I have decided to try a new approach to my self-study of mathematics.
In reality, I have no time to read the actual lessons per chapter in my books. Honestly, time is very short. I go back to work on May 1 (overnight hours), which means sleeping in the daytime is needed. Have you ever tried sleeping in the daytime when your noisy neighbors are up and about without any concern for your rest time?

My New Self-study Plan

1. Study one topic at a time.

2. Learn each topic watching video clips on YouTube and then searching my textbooks or online websites for similar problems to work on.

3. Concentrate on one math concept per week. For example, if I decide to review factoring trinomials, then factoring trinomials will be my only topic to work on for one week. The following week, I dive into a new topic and so forth.

4. I will only study what math professors on YouTube decided to cover for any given course. It makes ZERO SENSE for me to dive into material not covered by textbook authors, which explains why I ask for help with so many DIFFERENT types of problems not explained in the textbooks.

5. Bob, what do you think about 1 to 4?

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The greatest truth about the Rapture is not its timing but it's reality.

Dr. David Jeremiah

Bob

Administrator

Registered: 2010-06-20

Posts: 10,279

Re: Express Total Area A

Hopefully that will suit.  Good luck!

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 687

Re: Express Total Area A

Hopefully that will suit.  Good luck!

Bob

I think this is a better idea. Did you read number 4? This is ludicrous, to say the least. Why waste your time with problems that Michael Sullivan did not bother to explain in the lesson part of Section 3.6? I have been doing this for years. No wonder I always fall behind in my self-study of mathematics.

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The greatest truth about the Rapture is not its timing but it's reality.

Dr. David Jeremiah