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Calculus Circumscribe help

4startimer

New member
Dec 9, 2012
1
I am having a rather difficult time figuring out these 4 problems. Could someone please help.
the images that are provided are of a rectangle circumscribed around a circle, and a rectangle inscribed within a circle.
1. what is the area of the smallest rectangle the can be circumscribed around the unit circle?
2. what is the area of the largest rectangle that can be inscribed within the unit circle?
3. approximate the area of the unit circle using an appropriate number of circumscribed rectangles if width 0.4 units.
4. approximate the area of the unit circle using an appropriate number of inscribed rectangles if width 0.4 units.


How would I go about setting this up. I am fairly lost.


Picture: View image: problem

There are more problems than are pictured, so I figure if I can find out how to do these first four, I can complete the rest. I will be posting my work as I finish it in order to confirm that I am doing everything correctly.
 

CaptainBlack

Well-known member
Jan 26, 2012
890
I am having a rather difficult time figuring out these 4 problems. Could someone please help.
the images that are provided are of a rectangle circumscribed around a circle, and a rectangle inscribed within a circle.
1. what is the area of the smallest rectangle the can be circumscribed around the unit circle?
A rectangle circumscribed around a circle is any rectangle such that all the points of the circle are on or inside the rectangle.

It is obvious that any circumscribed rectangle to a circle can be shrunk untill all four sides are tangent to the circle. This condition forces it to be a square. Thus we observe that the minimum area of any cirrcumscribed rectangle to a circle is greater than or equal to the area of a square with each side a tangent to the circle. Such a square has a side equal to the diameter, so is of area 4 (since the diameter of a unit circle is 2).

CB
 

CaptainBlack

Well-known member
Jan 26, 2012
890
2. what is the area of the largest rectangle that can be inscribed within the unit circle?
It is quite clear that a maximal area inscribed rectangle has its vertices on the circle, and that a diagonal of the rectangle is a diameter of the circle. If one side of the rectangle is \(x\) then the area of the rectangle is \(A(x)=x\sqrt(4-x^2)\).

Now the x that maximises the area is found in the usual manner, by finding the stationary points of A(x) ...

CB
 

CaptainBlack

Well-known member
Jan 26, 2012
890
3. approximate the area of the unit circle using an appropriate number of circumscribed rectangles if width 0.4 units.
4. approximate the area of the unit circle using an appropriate number of inscribed rectangles if width 0.4 units.
Is the wording of part 3 , there are no such rectangles.

The wording of part 4 is also wrong.

CB
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
You should also realize that a "rectangle circumscribing a circle" or a "rectangle inscribed in a circle" is a square!