1. ## optimization question

1) A rectangle study area is to be enclosed by a fence and divided into two equal parts, with a fence running along the division parallel to one of the sides. If the total area is 384ft^2, find the dimenstions of the study area that will minimize the total length of the fence. How much fence will be required?

2) Show that among all rectangles with a given perimeter, the square has the largest area.

3) How close does the curve y=1/x come to the origin?

thank you. Any help would be appreciated.

2. Originally Posted by inhae772

1) A rectangle study area is to be enclosed by a fence and divided into two equal parts, with a fence running along the division parallel to one of the sides. If the total area is 384ft^2, find the dimenstions of the study area that will minimize the total length of the fence. How much fence will be required?

2) Show that among all rectangles with a given perimeter, the square has the largest area.

3) How close does the curve y=1/x come to the origin?

thank you. Any help would be appreciated.
1. let $\displaystyle F$ = total length of fence

$\displaystyle F = 2L + 3W$

$\displaystyle LW = 384$ ... $\displaystyle L = \frac{384}{W}$

$\displaystyle F = 2\left(\frac{384}{W}\right) + 3W$

find $\displaystyle \frac{dF}{dW}$ and minimize

2. $\displaystyle P = 2L + 2W$ , $\displaystyle P$ is a fixed constant

$\displaystyle L = \frac{P-2W}{2}$

$\displaystyle A = LW$

$\displaystyle A = \frac{P-2W}{2} \cdot W$

find $\displaystyle \frac{dA}{dW}$ and maximize

3. point on the curve ... $\displaystyle \left(x , \frac{1}{x}\right)$

origin $\displaystyle (0,0)$

distance formula ...

$\displaystyle D = \sqrt{(x-0)^2 + \left(\frac{1}{x} - 0\right)^2}$

$\displaystyle D = \sqrt{x^2 + \frac{1}{x^2}}$

shortcut ...
if you minimize $\displaystyle x^2 + \frac{1}{x^2}$ , then you also minimize the distance, $\displaystyle D$