Uni Maths Maximizing area of fencing

Does my solution seem suitable? I think that I may be thinking too simply and instead should create a differential equation but im not sure how I would do that.

Thank you in advance.

Does my solution seem suitable? I think that I may be thinking too simply and instead should create a differential equation but im not sure how I would do that.

Thank you in advance.

I'm studying chemistry so take this with a grain of salt, but surely:

Assuming you make a rectangle shape using the river as one side;
2X + Y = 1000 (the 1000 m of fencing). Y is the other side, which is equal to the side of the river (not included in the equation as no fencing is required for the river side).
The is an equation for the open perimeter.
The equation for the area would be A=XY (X multiplied by Y).
Rearrange the first equation to Y= 1000 - 2X
Sub this into the area equation:
A = X(1000 - 2X) thus A = 1000X -2X^2
Differentiate dA/dX to find the value of X at the turning point, which will give the maximum area.
We get 1000 - 4X =0. X cannot be a negative length thus X = 250 metres.
Sub into original equation; 500 + Y = 1000. Thus Y = 500m.
So you have two sides of fencing perpendicular to the river, both of length 250 meters.
Then one side of fencing at the end parallel to the river (connecting the 250 metre sides together) of length 500 meters. Maximum area = 500 * 250 = 125000 metres squared

How do you know the shape should be a semicircle? (It's correct that it's a semicircle, but I don't think it's obvious).

[This is known as Dido's problem, if you want to Google it].

Using a circle with radius 318 i get the area of the semi circle to be 158835 m^2 which is bigger than yours

Ah I see. No problem, maybe instead set up an equation for the radius and another for the area of the semicircle, then differentiate to find radius that will give the maximum area ?

We were given a hint at the begging of the lecture that a circle will give you the largest shape.

Does my solution seem suitable? I think that I may be thinking too simply and instead should create a differential equation but im not sure how I would do that.

Thank you in advance.

Area is πr^2, so obviously you just choose the largest possible radius (if you can assume the shape is a semi-circle). There's no "trade-off" where increasing the radius "costs" you area, so differentiating accomplishes nothing.


Since the best shape is a semi-circle, not a circle, I'm not sure what to say to this.

Finding the optimal shape is an isoperimetric problem. You generally use calculus of variations to solve it (Euler-Lagrange and all that).

If the last paragraph meant nothing to you, then yeah, just assume a semi-circle and your first post should be fine.

If you recognized the terms, then I think you're expect to prove a semi-circle is the optimal shape I'm afraid.

must you spell maximise with a z? ghastly.