A rectangular lot is bordered on one side by a stream and on the other three sides by 360m of fencing. Find the dimensions of the lot in order to maximize area.

(Also, does it make sense if the answer I got is 180m width, 90m length?)

So, we will assume there is no need for a fence along the stream.

let the side parallel to the stream be y m
let each of the other two sides be x m
so 2x + y = 360
y = 360-2x

area = xy = x(360-2x)
= -2x^2 + 360x

the vertex of this parabola produces a maximum
the x of the vertex is -b/(2a) = -360/-4 = 90
if x = 90 , then y = 360-2(90) = 180

You are correct

To find the dimensions of the rectangular lot that maximize the area, we can use the concept of calculus, specifically optimization. The problem gives us the total length of the fencing, which is 360m, and we need to determine the width and length of the lot.

Let's assume the width of the lot is "w" and the length is "l". We know that one side of the lot is bordered by a stream, so we have two lengths and one width to enclose. Therefore, the total length of fencing used is:

2l + w = 360

Now, we need an equation for the area of the lot in terms of "w" and "l". The area of a rectangle is given by:

Area = length × width
A = l * w

Now, we have two equations:
Eq1: 2l + w = 360
Eq2: A = l * w

To maximize the area, we need to solve these equations simultaneously. We can rewrite Eq1 as:

w = 360 - 2l

Substituting this value of "w" into Eq2, we get:

A = l * (360 - 2l)
A = 360l - 2l^2

To find the dimensions that maximize the area, we now need to find the critical points of this quadratic function. In this case, we'll use calculus to find the derivative of "A" with respect to "l" and set it equal to zero:

A' = 360 - 4l

Setting A' = 0:

360 - 4l = 0
4l = 360
l = 90

We have found that l = 90. Substituting this value back into Eq1, we can solve for "w":

2(90) + w = 360
180 + w = 360
w = 360 - 180
w = 180

So, the dimensions that maximize the area of the lot are a length of 90m and a width of 180m.

Therefore, your answer of 180m width and 90m length is correct.