MHB How Do Constraints and Graphs Relate to Concentric Circles?

[mathland]

Two concentric circles have radii x and y, where y > x. The area between the circles is at least 10 square units.

(a) Write a system of inequalities that describes the constraints on the circles.

What does the word CONSTRAINTS mean here?

(b) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in
the context of the problem.

What exactly is part (b) asking for?

 

[topsquark]

A constraint is a restriction on the equation or the values given. For example, we can constrain the graph of [math]y = x^2[/math], which normally acts on all real numbers x, to just be for all x > 0.

If you are up on function notation, [math]y = x^2[/math] is the set [math]\{ y(x) = x^2 | x \in \mathbb{R} \}[/math]. Everything after the | is some kind of constraint.

-Dan

 

[mathland]

A constraint is a restriction on the equation or the values given. For example, we can constrain the graph of [math]y = x^2[/math], which normally acts on all real numbers x, to just be for all x > 0.

If you are up on function notation, [math]y = x^2[/math] is the set [math]\{ y(x) = x^2 | x \in \mathbb{R} \}[/math]. Everything after the | is some kind of constraint.

-Dan

Let me work on this a bit more.

 

[HOI]

Two concentric circles have radii x and y with y> x. Do you know what "concentric" means? What is the area of the circle with radius x? What is the area of the circle with radius y? What is the area of the region between the circles?

 

[mathland]

Two concentric circles have radii x and y with y> x. Do you know what "concentric" means? What is the area of the circle with radius x? What is the area of the circle with radius y? What is the area of the region between the circles?

a) y > x;

x > 0;

pi(y^2 - x^2) >= 10.

(b) No idea what is being requested.

 

[HOI]

You quoted my questions but didn't actually answer them.
I presume you mean that you do know that "concentric circles" are circles that have the same center and so one, the one with the smaller radius, is inside the other.

A circle with radius "x" has area $\pi x^2$, a circle with radius "y" has area $\pi y^2$ and, since y> x, the area between them is $\pi y^2- \pi x^2= \pi(y^2- x^2)$. We are told that the area between the two circles is "at least 10 square units" so the "constraint" is, as you say, $\pi(y^2-x^2)\ge 10$.

(b) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in
the context of the problem.

I am not sure what "line" this is asking about. The best I can do is say that the graph of $\pi(x^2- y^2)= 10$, or $x^2- y^2= \frac{10}{\pi}$ is a hyperbola with axes along the coordinate axes and vertices at $\left(\pm\sqrt{\frac{10}{\pi}}, 0\right)$. The inequality $\pi(x^2- y^2)\ge 10$ is satisfied to the right and left of that hyperbola so the hyperbola itself is the boundary.
("Identify the graph of the line in relation to the boundary of the inequality" sees to me awkward English grammar. I would have said "identify the graph forming the boundary of the inequality". Also the use of the word "line" makes me wonder if they do not mean the asymptotes of the hyperbola, the lines y= x and y= -x, but that seems unlikely. )

 

[mathland]

You quoted my questions but didn't actually answer them.
I presume you mean that you do know that "concentric circles" are circles that have the same center and so one, the one with the smaller radius, is inside the other.

A circle with radius "x" has area $\pi x^2$, a circle with radius "y" has area $\pi y^2$ and, since y> x, the area between them is $\pi y^2- \pi x^2= \pi(y^2- x^2)$. We are told that the area between the two circles is "at least 10 square units" so the "constraint" is, as you say, $\pi(y^2-x^2)\ge 10$.I am not sure what "line" this is asking about. The best I can do is say that the graph of $\pi(x^2- y^2)= 10$, or $x^2- y^2= \frac{10}{\pi}$ is a hyperbola with axes along the coordinate axes and vertices at $\left(\pm\sqrt{\frac{10}{\pi}}, 0\right)$. The inequality $\pi(x^2- y^2)\ge 10$ is satisfied to the right and left of that hyperbola so the hyperbola itself is the boundary.
("Identify the graph of the line in relation to the boundary of the inequality" sees to me awkward English grammar. I would have said "identify the graph forming the boundary of the inequality". Also the use of the word "line" makes me wonder if they do not mean the asymptotes of the hyperbola, the lines y= x and y= -x, but that seems unlikely. )

Concentric circles are circles within circles.

 

[mathland]

You quoted my questions but didn't actually answer them.
I presume you mean that you do know that "concentric circles" are circles that have the same center and so one, the one with the smaller radius, is inside the other.

A circle with radius "x" has area $\pi x^2$, a circle with radius "y" has area $\pi y^2$ and, since y> x, the area between them is $\pi y^2- \pi x^2= \pi(y^2- x^2)$. We are told that the area between the two circles is "at least 10 square units" so the "constraint" is, as you say, $\pi(y^2-x^2)\ge 10$.I am not sure what "line" this is asking about. The best I can do is say that the graph of $\pi(x^2- y^2)= 10$, or $x^2- y^2= \frac{10}{\pi}$ is a hyperbola with axes along the coordinate axes and vertices at $\left(\pm\sqrt{\frac{10}{\pi}}, 0\right)$. The inequality $\pi(x^2- y^2)\ge 10$ is satisfied to the right and left of that hyperbola so the hyperbola itself is the boundary.
("Identify the graph of the line in relation to the boundary of the inequality" sees to me awkward English grammar. I would have said "identify the graph forming the boundary of the inequality". Also the use of the word "line" makes me wonder if they do not mean the asymptotes of the hyperbola, the lines y= x and y= -x, but that seems unlikely. )

You said:

"Identify the graph of the line in relation to the boundary of the inequality" sees to me awkward English grammar. I would have said "identify the graph forming the boundary of the inequality". Also the use of the word "line" makes me wonder if they do not mean the asymptotes of the hyperbola, the lines y= x and y= -x, but that seems unlikely."

The English found in most math and physics textbooks is awkward in so many ways. In fact, this is the main reason why I struggle with applications.

 

[HOI]

Concentric circles are circles within circles.

No, not all "circles within circles" are concentric. Concentric circle are circles that have the same center. It follows that one is inside the other. The circle $(x- 1)^2+ y^2= 1$ is "within" $x^2+ y^2= 9$ but they are not "concentric".