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Calculus/largest inscribed triangle

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Expert: Alon Mandes - 1/1/2010


Question
Points A and B lie at the ends of a diameter of a unit circle and point C lies on the circumference.Is it true that the area of triangle ABC is largest when the triangle is isosceles?How do you know?

Answer
The angle ACB is 90 degrees . The sides of the right triangle are :
AB=d (Diameter)
AC=y
BC=x
According to Pythagoras : x^2+y^2=d^2 . Therefore y=(d^2-x^2)^(1/2) .
Area S=(1/2)xy=(1/2)[x(d^2-x^2)^(1/2)] .
S'=(1/2)[(d^2-x^2)^(1/2)+x(-x)(d^2-x^2)^(-1/2) .
S'=0 --> (d^2-x^2)^(1/2)=x^2/(d^2-x^2)^(1/2) .
d^2-x^2=x^2
d^2=2x^2 --> x=0.707d
Let's calculate y : y=(d^2-x^2)^(1/2)=(d^2-0.5d^2)^(1/2)=(0.5d^2)^(1/2)=0.707d .
Hence x=y it's an isosceles triangle . Q.E.D

Alon.  

Calculus

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Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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