Thread: Calculus

Hello Forumites,
A farmer wishes to fence a field bordering a straight stream with 1000 yd. of fencing material. It is not necessary to fence the side bordering the stream. Now, i want to find what is the maximum area of a rectangular field that can be fenced in this way?

Solution. I don't have any idea to solve this problem. I know the area of rectangle=length*width. Now, for maxima, first derivative=0 and second derivative <0. Answer provided to me is 125000 sq.yd. If any forumite know the answer, he may reply with it.

Assume all units are yds or sq yds as appropriate.

$l + 2w = 1000$

$A = l w$

$A = (1000-2w)w = 1000w - 2w^2$

$\dfrac{dA}{dw} = 1000 - 4w$

solve for $\dfrac{dA}{dw}=0$

$4w = 1000$

$w = 250$

$l = 1000 - 2(250) = 500$

$A = (250)(500) = 125000$

I would let $\displaystyle x$ be the length of the sides perpendicular to the stream and $\displaystyle y$ be the length of the side parallel to the stream. Let the total amount of fence be $\displaystyle F$.

The area $\displaystyle A$ of the enclosed rectangle gives us the objective function:

$\displaystyle A(x,y)=xy$

Subject to the constraint:

$\displaystyle g(x,y)=2x+y-F=0$

Using Lagrange multipliers, there arises the system:

$\displaystyle y=\lambda(2)$

$\displaystyle x=\lambda(1)$

This implies:

$\displaystyle y=2x$

Substituting into the constraint, we obtain:

$\displaystyle 4x=F\implies x=\frac{F}{4},\,y=\frac{F}{2}$

We then find:

$\displaystyle A\left(\frac{F}{4},\frac{F}{2}\right)=\frac{F^2}{8 }$

As $\displaystyle F(0,F)=F\left(\frac{F}{2},0\right)=0$, we may then conclude:

$\displaystyle A_{\max}=\frac{F^2}{8}$

Using the given value for $\displaystyle F$, we find:

$\displaystyle A_{\max}=\frac{(1000\text{ yd})^2}{8}=125000\text{ yd}^2$

I know you are looking for a solution using functions of one variable, but this should easily give you enough to do so. Solve the constraint for one of the variable, and then you can express the area in terms of that variable, and the parameter $\displaystyle F$.

On a timed test, with a few seconds to go,
easy for an unsuspecting(!) student to reason
"since square produces maximum area,
then 333 1/3...."

Originally Posted by DenisB

On a timed test, with a few seconds to go,
easy for an unsuspecting(!) student to reason
"since square produces maximum area,
then 333 1/3...."

How would an unsuspecting student come up with that!?

Originally Posted by greg1313

How would an unsuspecting student come up with that!?

That's a good question. Perhaps we should ask my students who came up with that back when I taught Calc I.

Originally Posted by DenisB

...333 1/3...."

Isn't that an old record player setting?

-Dan

Originally Posted by topsquark

Isn't that an old record player setting?

-Dan

It was 33 1/3. I recall 16, 33 1/3, 45 and 78. My dad had some 78's, but all I ever had were 33 1/3 and 45.

Originally Posted by MarkFL

It was 33 1/3. I recall 16, 33 1/3, 45 and 78. My dad had some 78's, but all I ever had were 33 1/3 and 45.

I know. I was just messing around. I even remember 8 tracks. And my ex-father-in-law had a reel to reel.

-Dan

I remember one of these:
https://www.ebay.com/bhp/victrola-crank

My dad used to listen to this on it:

Jambalaya (On the Bayou)

Jambalaya, crawfish pie and fillet gumbo
For tonight, I'm-a gonna see my ma cher a mi-o
Pick guitar, fill fruit jar and be gay-o
Son of a gun, we'll have big fun on the bayou…

Originally Posted by topsquark

I know. I was just messing around. I even remember 8 tracks. And my ex-father-in-law had a reel to reel.

-Dan

I had friends with 8-tracks in their cars...I was a cassette lad myself during that period. One of my uncles, a real audiophile, had a reel to reel system. I still think analog music sounds more vibrant than digital.

Originally Posted by DenisB

I remember one of these:
https://www.ebay.com/bhp/victrola-crank

My dad used to listen to this on it:

Jambalaya (On the Bayou)

Jambalaya, crawfish pie and fillet gumbo
For tonight, I'm-a gonna see my ma cher a mi-o
Pick guitar, fill fruit jar and be gay-o
Son of a gun, we'll have big fun on the bayou…

My dad was a HUGE Hank fan...I know all his songs by heart.

Originally Posted by DenisB

I remember one of these:
https://www.ebay.com/bhp/victrola-crank

My dad used to listen to this on it:

Jambalaya (On the Bayou)

Jambalaya, crawfish pie and fillet gumbo
For tonight, I'm-a gonna see my ma cher a mi-o
Pick guitar, fill fruit jar and be gay-o
Son of a gun, we'll have big fun on the bayou…

How about
Chicken in the bread pan peckin' out dough,
Granny does your dog bite? No, child, no.
(That's the Devil Went Down to Georgia version anyway.)

-Dan

...and why did you guys "pick on my 333 1/3" anyway ?

Originally Posted by DenisB

...and why did you guys "pick on my 333 1/3" anyway ?

I have to have a reason??

-Dan