maximum area

[chris2005]

The lifeguard at a public beach has 400 m of rope available to lay out a rectangle restricted swimming area using the straight shorline as one side of the rectangle.

a) if she wants to maximize the swimming area, what will the dimensions of the rectangle be?

b) To ensure safety of swimmers, she decides that nobody should be more then 50 m from shore. What should the dimensions of the swimming area be with this added restriction?

 

[chris2005]

its a very complicated question

 

[galactus]

It's not that bad, Chris. There's been tougher.

Since the lifeguard only needs 3 sides roped in (because the shoreline is one

side), and she has 400 meters of rope, she can set up the perimeter

as such:

\(\displaystyle 2y+x=400\)

The area is given by \(\displaystyle xy=A\)

Solve the top equation for, say, x:

\(\displaystyle 400-2y=x\)

Sub into area equation:

\(\displaystyle (400-2y)y=A\)

Differentiate:

\(\displaystyle \frac{dA}{dy}=400-4y\)

Set to 0 and solve for y:

\(\displaystyle 400-4y=0\)

\(\displaystyle y=100\)

If \(\displaystyle y=100\), then \(\displaystyle x=200\)

Area=\(\displaystyle (100)(200)=20,000 m^{2}\)

Now, can you set up the second half of the problem by incorporating the 50 m restriction?.