Maximizing the Area of a Pentagon with Fixed Perimeter

[joemama69]

Homework Statement

a pentagon is formed by placing an isosceles triangle on a rectangle. If the pentagon has a fixed perimeter P, determine the lengths of the sides of the pentagon that maximize the area of the pentagon

Homework Equations

The Attempt at a Solution

not sure where to start since it is a irregular pentagon

 

[tiny-tim]

a pentagon is formed by placing an isosceles triangle on a rectangle. If the pentagon has a fixed perimeter P, determine the lengths of the sides of the pentagon that maximize the area of the pentagon
…
not sure where to start since it is a irregular pentagon

What's the difficulty?

Call the lengths of the sides a b and c, then add the area of the rectangle to that of the triangle, and maximise.

 

[FedEx]

What's the difficulty?

Call the lengths of the sides a b and c, then add the area of the rectangle to that of the triangle, and maximise.

I think there is a problem. Cause we are dealing with three variables but we have just two equtions 1) 2a + 2b + c =P and 2) The one which tells about the total area. By normal convention we diff equation number 2. But we have three variables out of which we are able to eliminate just one.

Or else we could convert the equation number 2) in a quadratic of b. I am not totally sure whether that works or not

 

[Dick]

I think there is a problem. Cause we are dealing with three variables but we have just two equtions 1) 2a + 2b + c =P and 2) The one which tells about the total area. By normal convention we diff equation number 2. But we have three variables out of which we are able to eliminate just one.

Or else we could convert the equation number 2) in a quadratic of b. I am not totally sure whether that works or not

Why do you think there is a problem? The perimeter constraint eliminates one variable. Now you have an equation for the area in terms of two variables. You 'diff' equation 2 with respect to each one of the two remaining variables and set them equal to zero. That's two equations in two unknowns. Where is problem?

 

[FedEx]

Why do you think there is a problem? The perimeter constraint eliminates one variable. Now you have an equation for the area in terms of two variables. You 'diff' equation 2 with respect to each one of the two remaining variables and set them equal to zero. That's two equations in two unknowns. Where is problem?

Are you telling me to differentiate it partially?

Well course.

Well the thing is that we haven't been taught with partial differentiation. there's not much to it in undergraduate math... but still it does make a diff if you have learned it formally

 

[joemama69]

heres what i got so far

A(1) = ab
A(2) = .5a$$\sqrt{c^2 - (a/2)^2}$$

Total A = A(1) + A(2) = ab + .5a$$\sqrt{c^2 - (a/2)^2}$$

P = a + 2b + 2c

"Why do you think there is a problem? The perimeter constraint eliminates one variable."

by that do mean solving for let's say a = P - 2b - 2c
and then i would plug it into the other equation

A = (P - 2b - 2c)b + .5(P - 2b - 2c) $$\sqrt{c^2 - (P - 2b - 2c) /2^2}$$

"Now you have an equation for the area in terms of two variables"

seems like there's three to me, P,b,c

"You 'diff' equation 2 with respect to each one of the two remaining variables and set them equal to zero."

which variables do i differentiate in terms of

 

[Dick]

P isn't a variable. It's 'fixed'. Your final answers for a, b and c will need to be given in terms of P.

 

[FedEx]

P isn't a variable. It's 'fixed'. Your final answers for a, b and c will need to be given in terms of P.

Precisely. So diff P w.r.t to any variable would give you zero

 

[joemama69]

ok

A = Pb - 2b² - 2bc + (P/2 - b - c)(2c² - P²/4 + Pc + Pb + b² + 2bc)^1/2

A_b = P - 4b + 2c + 1/2(p/2 - b - c)(P + 2b + 2c)(2c² - p²/4 + Pc + Pb + b² + 2bc)^-1/2 - (2c² - p²/4 + Pc + Pb + b² + 2bc)^1/2

A_b = P - 4b + 2c + (P²/8 - b² - 2c² - 3bc - Pc/2)/(2c² - P²/4 + Pc + Pb + b² + 2bc)^1/2 - (2c² - P²/4 + Pc + Pb + b² + 2bc)^1/2

So that is the Partial of A with respect to b. Based on what u told me i will also have to differentiate A with respect to c. I should set them both equal to 0, find all critical points, and then figure out which one is the maximum.

Have i done it right so far. Its quite messy, is there an easier way or do i just have to deal w/ it

 

[Dick]

I think it's somewhat messy however you do it. But doing it that way I definitely would have eliminated b instead of a. The easiest way is probably to put P in as a Lagrange multiplier, if you've learned that.

 

[joemama69]

Nope we never learned that. this is pretty messy, if you think its work it do u mind walkin me through it

BTW, i solved bor b instead, much simpler

A'(a,c) = P/2 - 2a - c + (2ac - a^2/2)/(4$$\sqrt{c^2 - a^2/4}$$) + $$\sqrt{c^2 -a^2/4}$$/2

 

[Dick]

Nope we never learned that. this is pretty messy, if you think its work it do u mind walkin me through it

BTW, i solved bor b instead, much simpler

A'(a,c) = P/2 - 2a - c + (2ac - a^2/2)/(4$$\sqrt{c^2 - a^2/4}$$) + $$\sqrt{c^2 -a^2/4}$$/2

I would start with the d/dc equation rather than the d/da. It's MUCH simpler. It's pretty easy to find the relation between a and c and eliminate another variable in d/da.

 

[joemama69]

I assume that means the partial of A with respect to c

A_c = -a + ac / (2$$\sqrt{c^2 - a^2/4}$$) = 0

a = ac / (2$$\sqrt{c^2 - a^2/4}$$) Square both sides to solve for a

a² = a²c² / (4c² - a²)

4a²c² - a⁴ = a²c²

a² = 4c² - c² therefore a = c$$\sqrt{3}$$

0 = -c$$\sqrt{3}$$ + (c²$$\sqrt{3}$$) / (2(c² - 3c²/4)^1/2)

0 = -c$$\sqrt{3}$$ + (c²$$\sqrt{3}$$) / 2(c²/4)^1/2

0 = -c$$\sqrt{3}$$ + (c²$$\sqrt{3}$$) / c

0 = -c$$\sqrt{3}$$ + c$$\sqrt{3}$$

something aint right. where i go wrong

 

[Dick]

I'll agree with a=sqrt(3)*c. Beyond that I think you should learn to check you own work. Would you try it again and if it doesn't work post clearly what you got for dA/da. I agree with parts of what you got, but I didn't get the same form for the whole thing. I think you are capable of finishing this on your own. That's a compliment.

 

[joemama69]

Yup got the same thing

Ok so

A_c = -a + (ac)/(2$$\sqrt{c^2 - (a^2)/4}$$) = 0, a = $$\sqrt{3}$$c

0 = -$$\sqrt{3}$$c + ($$\sqrt{3}$$c²)/(2$$\sqrt{c^2 - (3c^2)/4}$$ )

0 = -$$\sqrt{3}$$c + ($$\sqrt{3}$$c²)/(2$$\sqrt{(c^2)/4$$)

0 = -$$\sqrt{3}$$c + ($$\sqrt{3}$$c²)/((2c)/2)

0 = -$$\sqrt{3}$$c + ($$\sqrt{3}$$c²)/c

0 = -$$\sqrt{3}$$c + ($$\sqrt{3}$$c


Did i plug my a into the wrong equation

 

[Dick]

It looks to me like you are plugging a=sqrt(3)*c back into the same equation you used to derive that relation. Plug it into dA/da, not dA/dc.

 

[joemama69]

that thought crossed my head

ok so ..

A_a = P/2 - a - c - a²/(8$$\sqrt{c^2 - (a^2)/4}$$) + ($$\sqrt{c^2 - (a^2)/4}$$)/2 = 0, plug in a = $$\sqrt{3}$$c

0 = P/2 - $$\sqrt{3}$$c - c - 3c²/(8$$\sqrt{c^2 - (3c^2)/4}$$) + ($$\sqrt{c^2 - (3c^2)/4}$$)/2

0 = P/2 - $$\sqrt{3}$$c - c - 3c²/(4c) + c/4

0 = P/2 - $$\sqrt{3}$$c - c - 3c/4 + c/4

P = c(2$$\sqrt{3}$$ + 7/2)

c = P/(2$$\sqrt{3}$$ + 7/2)

 

[Dick]

Fine, except you made an arithmetic error. In problems like these you can have several choices of variables to eliminate, and several choices of which equation and variable to solve for first. Try and look ahead and figure out which choice will make your life easier.

 

[joemama69]

Where is my error @, i can't find it

 

[Dick]

Where is my error @, i can't find it

Between the third line from the end and the next one.

 

[joemama69]

ok found it

so c = P/(2$$\sqrt{3}$$+3/2)

and a = $$\sqrt{3}$$c = $$\sqrt{3}$$P/(2$$\sqrt{3}$$+3/2)

what do i do from here, I am lost

 

[Dick]

ok found it

so c = P/(2$$\sqrt{3}$$+3/2)

and a = $$\sqrt{3}$$c = $$\sqrt{3}$$P/(2$$\sqrt{3}$$+3/2)

what do i do from here, I am lost

Uh, maybe find b? That 3/2 still isn't right.

 

[joemama69]

sorry stupid question

P = a + 2b + 2c

P = $$\sqrt{3}$$P/(2$$\sqrt{3}$$ + 3/2) + 2b + 2P/(2$$\sqrt{3}$$ + 3/2)

2b = P - $$\sqrt{3}$$P/(2$$\sqrt{3}$$ + 3/2) - 2P/(2$$\sqrt{3}$$ + 3/2)

b = P/2[1 - ($$\sqrt{3}$$-2)/(2$$\sqrt{3}$$ + 3/2)]

 

[joemama69]

0 = P/2 - $$\sqrt{3}$$c - c - (3c/4) + c/4 move everything over except the p

P/2 = $$\sqrt{3}$$c + c + (3c/4) - c/4

P/2 = c($$\sqrt{3}$$ + 1 + (3/4) - 1/4)

P/2 = c($$\sqrt{3}$$ + 1 + (3/4) - 1/4

P/2 = c($$\sqrt{3}$$ + 3/2)

c = P/(2(($$\sqrt{3}$$ + 3/2)


c = P/(2$$\sqrt{3}$$ + 3) gotcha

that makes b = (P/2)/[1 - ($$\sqrt{3}$$ -2)/(2$$\sqrt{3}$$ + 3)]