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mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Semicircle Inscribed In Rectangle

A semicircle of radius r is Inscribed in a rectangle so that the diameter of the semicircle is the length of the rectangle.

A. Express the area A of the rectangle as a function of the radius r of the semicircle.

B. Express the perimeter p of the rectangle as a function of r.

Part A

Length of rectangle = diameter of semicircle.

We want A(r).

Area of semicircle = (1/2)(pi)•r^2.

Stuck here....

Part B

We want p(r).

Perimeter of rectangle = 2L + 2W.

p(r) = 2r + 2(2r)

p(r) = 2r + 4r

p(r) = 6r

You say?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,637

Re: Semicircle Inscribed In Rectangle

it doesn't fully make it clear but I think 'inscribed' here means the semicircle touches the rectangle at the top as well as having it's diameter in common with the long side of the rectangle. That means the rectangle is 2r by r.  Without that none of the question is do-able.

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: 1,053

Re: Semicircle Inscribed In Rectangle

it doesn't fully make it clear but I think 'inscribed' here means the semicircle touches the rectangle at the top as well as having it's diameter in common with the long side of the rectangle. That means the rectangle is 2r by r.  Without that none of the question is do-able.

Bob

According to the textbook, the answer for part A is 2r^2.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,637

Re: Semicircle Inscribed In Rectangle

Yes, I got that already. 

Draw a semicircle, flat diameter at the bottom. That is also the long side of the rectangle. Draw up from the ends and across at the top of the semicircle so you make two shorter sides and the final side is a tangent to the circle.

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: 1,053

Re: Semicircle Inscribed In Rectangle

Yes, I got that already. 

Draw a semicircle, flat diameter at the bottom. That is also the long side of the rectangle. Draw up from the ends and across at the top of the semicircle so you make two shorter sides and the final side is a tangent to the circle.

Bob

Perfect. I will play around with this section by returning to the start of 3.6 in the Michael Sullivan College Algebra Edition 9 textbook.