# Thread: optimization problems and a related rates problem

1. ## optimization problems and a related rates problem

1) A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for $3 a foot, while the remaining two sides will be standard fencing selling for$2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000? 2) Find the point on the graph of y= sqrt(x+1) that is closest to (8,0) 3) A 25 foot long ladder is leaning against the wall of a house. If the base of the ladder is pulled away from the wall at the rate of 2 feet per second, at what rate is the area formed by the side of the house, the ladder and the ground changing at the moment the base of the ladder is 15 feet from the wall? i honestly dont know where to even begin with #2, as for #1, my answers don't make sense so i'm definately doing it wrong. #3... i have an idea but it's not working out the way i planned. can someone please help please and thank you 2. Originally Posted by vtong 1) A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for$3 a foot, while the remaining two sides will be standard fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of$6000?
let the lengths of the sides be $x$ and $y$ ( $x$ the length of the sides with heavy duty fencing)

Then cost:

$3\times 2 \times x + 2 \times 2 \times y=6000.$

Area:

$A=xy=x \frac{6000-6x}{4}$

Now to find the $x$ that gives maximum area you differentiate $A$ with respect to x and set this to zero and solve for the critical point/s etc.

Then use this area maximising $x$ to find the corresponding $y$

CB

3. Originally Posted by vtong

2) Find the point on the graph of y= sqrt(x+1) that is closest to (8,0)
If the point $(x,y)$ lies on the graph then its square distance from $(8,0)$ is:

$D=d^2=(8-x)^2+y^2=(8-x)^2-(x+1)$

Now find the $x$ that minimises $D$ etc.

CB