Download PROBLEM 16 : What angle between two edges of

PROBLEM 16 : What angle
between two edges of length 3 will result in an
isosceles triangle with the largest area ? (See diagram.)
SOLUTION
Write the area of the given isosceles triangle as a function of
. Let variable x
be the length of the base and variable y the height of the triangle, and
consider angle
. Write each of x and y as functions of .
It follows from basic trigonometry that
so that
(Equation 1 )
,
and
so that
(Equation 2 )
We wish to MAXIMIZE the AREA of the isosceles triangle
A = (1/2) (length of base) (height) = (1/2) xy .
Before we differentiate, use Equations 1 and 2 to rewrite the right-hand side as
a function of only. Then
A = (1/2) xy
.
Now differentiate this equation using the product rule and chain rule, getting
(Factor out (9/2) and simplify the expression.)
=0,
so that
and
.
It follows algebraically (Why ?) that
so that from basic trigonometry we get
or
,
or
.
and hence
Because
measures an angle in a triangle, it is logical to assume that
. Thus,
. See the adjoining sign chart for A' .
If
radians = 90 degrees,
then
A = 9/2
is the largest possible area for the triangle.