Assign variables to the unknowns.
Let's assign variables to the unknowns. Let
denote the total area,
the length of the sides of the triangle, and
the length of the sides of the square.
Find the height of the triangle.
Using the Pythagorean theorem, we find that the height of the triangle is
Find the primary equation.
The total area of the equilateral triangle and square has to be minimized, so the primary equation is
Find the secondary equation.
Given the total perimeter of the equilateral triangle and square is 10, so the secondary equation is 4
Solve the secondary equation for
Solving the secondary equation for
, we find that
= (10 – 3
Express the primary equation in terms of
Let's express the primary equation in terms of
. This is
, we get
Find the critical points.
Solving the equation
′ = 0, we find that the solution is
is minimized for this value of
We have to find the second derivative of
to verify. The second derivative is
" is greater than zero, the Second Derivative test tells us that
has a minimum for the value of
Find the lengths of the sides of the square.
The corresponding lengths of the sides of the square are
Simplifying, we get
Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC.
4.9/5.0 Satisfaction Rating over the last 100,000 sessions. As of 4/27/18.
*See complete details for Better Score Guarantee.
Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors.
Award-Winning claim based on CBS Local and Houston Press awards.
Varsity Tutors does not have affiliation with universities mentioned on its website.
Varsity Tutors connects learners with experts. Instructors are independent contractors who tailor their services to each client, using their own style,
methods and materials.