What are the dimensions of a cross section that minimizes perimeter?

[xcolleenx]

Homework Statement

This is from the book Calculus Single Variable, 4th edition:
The cross-section of a tunnel is a rectangle of height h surmounted by a semicircular roof section of radius <i>r</i>. If the cross-sectional area is A, determine the dimensions of the cross section which minimize the perimeter.

Homework Equations

I guess I am just lost on how to even approach this problem. I know you will need to use the formula for perimeter but do you need to even use the semicircle? Is the A referring to the rectangular section?

To see a picture of this: http://www.wiley.com/college/sc/hugheshallett/chap4.pdf and it is question number 25 in section 4.5. (Problems)

The Attempt at a Solution

I would start out by calling the height h and then the base of the rectangle d or diameter because the base is the same as the diameter of the circle on top. I would then note that the area of the rectangle is h times d and then i think i would need the formula for area or circumfrence of a circle so that i could fill "d" in with something. I guess i know what i would need to use, i think, i just don't know how to use it.

 

[D H]

The question is worded a bit poorly. My take is that the semicircular roof section is a part of the cross-section and hence contributes to the cross-section area and perimeter.

The way to attack this problem doesn't really depend on whether or not the semicircular section counts as part of the cross section. The cross-section area A is a function of the height h and the radius r. Since the area is a constant, this means you only have one independent variable. Pick one. Find the derivative of the perimeter with respect to this one variable and set to zero.

 

[Dick]

The perimeter P is the sum of the length of the semicircle and the sides and bottom of the rectangle. The area A is the sum of the area of the rectangle and the area of the semicircle. You can find formulas for all of these things easily. Now A is fixed. Use A to solve for one of the variables h or r and substitute it into P. Now minimize P.

 

[xcolleenx]

Thank you for your help. I have been working on this problem and I am still having trouble. I will do the best to type out the work I have done so far.

P=4r+2h
A=2rh+((pi)(r^2)/2)

I then solved the area formula for h and came up with:
h=(A-(pi)(r^2)/4r)
And then plugged this in for h in the perimeter formula.
Then i found the derivative of the perimeter formula and came up with:
P'=-16(pi)r^2-8A+8(pi)r+4
Then i need to find where this is equal to zero, correct? Here is where I am having trouble. Could you tell me if I am doing this right or did I mess up somewhere? Thank you very much.

 

[D H]

You have the right idea but made some mistakes in implementing it. You don't have the perimeter right. You have the perimeter of the rectangle, not the cross section. You do have the area of the cross section correct, but you made a mistake in computing the height as a function of the radius. Finally, the derivative isn't right.

 

[xcolleenx]

Okay. I am figuring this out slowly but I am. So now I have for P= 2h+2r+(pi)r and for A=2rh+1/2(pi)r^2. I solved for h and got h=(a-1/2(pi)r^2)/(2r). After filling that back into the perimeter formula I got P=((2a-(pi)r^2)/(2r))+2r+(pi)r. For the derivative, which is where I am having trouble, I got P'=((-2a)/(r^2))+2. Is this right or am I still on the wrong track? I think I am doing it right but I don't think my derivative is right.

 

[D H]

As all factors of [itex]\pi[/itex] have vanished from your derivative, you obviously have made some mistake. Try expanding and then simplifying your perimeter formula before you take the derivative.