- Consider a circle of radius with an inscribed isosceles triangle. If the angle at the apex of the isosceles triangle is , and , find the extreme values (maximum and minimum) of the perimeter of the triangle.
- Find the minimum value of the radius such that the circle of radius will cover every isosceles triangle with fixed perimeter .
- In the picture we draw in six radial lines, which divide the triangle into six congruent triangles, each with hypotenuse of length and legs of lengths and . As shown in the diagram, two of the edges of the inscribed isosceles triangle have lengths , and the third leg as length . Therefore, the perimeter is,
Taking the derivative,
Now, we must note that the extreme values of the function occur when the derivative is 0 or at an end point of the interval we are looking at; in this case, is the interval stipulated in the problem.
First, let’s set the derivative equal to zero and find the values of in the interval at which the derivative is 0.
Then, the only value of in the interval such that or is
Plugging this in to our expression for the perimeter of the isosceles triangle, we find,
Examining the sign of the derivative of the perimeter function as changes, we see that this is a maximum. Furthermore, we see that the minimum occurs at the end point, . The minimum perimeter is then
- First, we identify the value of at which the radius necessary to cover the isosceles triangle is at a maximum (the worst case scenario for covering the triangle, if you like). From part (a) we know the perimeter must satisfy the equation
Since is now a fixed constant we can write as a function of ,
Now, the value of is maximal when the denominator is minimal, so we are looking for a minimum of the function
Taking the derivative,
We set this equal to zero to find the critical points
From part (a) we know this implies or . Again, from part (a) we know this is minimal when . Since we are trying to minimize this we want the value .
Finally, there is a bit of a problem with since that corresponds to just a line, not really a triangle. We can either ignore that issue, and just plug in or (in what will amount to the same thing) we can find the minimal value of as the limit as :
where we can compute the limit by just evaluating at since the function is continuous.