I need to find the max size of these red rectangles in the attached picture.
I started off with a circle of 1ft radius, I found that the large square is 2ft^2
now I dont know how to maximise the area the red squares can be? please help
here is an easier way to tackle this, but it uses the exact same diagram and the calculations do not differ too much.
The first thing to imagine on the sketch is the height of the green rectangle going to zero. This means the top right hand corner of the rectangle goes down to point "p".
Here, the rectangle area is zero, since it's height is zero, though the base is a maximum.
It corresponds to angle A = 45 degrees (pink line with slope = 1).
Next, consider the point "p" moving to the point "z" with the top right corner of the rectangle attached to it.
At "z", the angle A is 90 degrees, the height is a maximum but the area of the rectangle is again zero, since the base is zero this time.
Somewhere between these extremes, the rectangle will reach a maximum area, as it starts from zero, increases to a maximum and falls to zero again.
To discover this maximum, we use the angle A, pivoted at the circle centre, since the rectangle is in a circle.
There is a little diagram to the left of the circle to help understand the calculations, where you can view the little triangles formed from the angle A.
The area of the rectangle is 2(e+c)s.
The distance from the circle centre to the bottom of the rectangle is
since the circle radius is 1 and
We require e, c and s in terms of the angle A.
From the leftmost right-angled triangle.......
Also, from the small top right triangle
Hence, we can continue with calculations until we have a compact formula for the rectangle area...
Substituting for c and s,
This is a more compact formula for rectangle area.
We need to find the maximum area of the rectangle.
To do that, we differentiate the area formula with respect to A and set the derivative equal to zero.
We use the product rule of differentiation for this.
This is a quadratic equation of the form
As we are looking for an angle between 45 and 90 degrees, we can eliminate the negative result.
This angle corresponds to maximum area.
Hence, maximum area is
The rectangle has the dimensions l and w and the area is
According to the attached sketch
To simplify the following calculations plug in r = 1. Differentiate a(x) wrt x and solve for x
You'll get 2 possible solutions and you have to determine which one belongs to the maximum.