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Calculus/maximum area of rectangle inscribed in a triangle

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Expert: Alon Mandes - 9/6/2009


Question
Give the largest rectangle inscribed in the triangle formed by line y=0. y=x and 3x+y=20 if one side of the rectangle lies on the x-axis

Answer

Drawing
Hello Erwin,
I attached an image that illustrate the drawing of the rectangle inscribed in the triangle.
We know that , the height of the point b will be : 20-3b . Thus, we obtain :
a=20-3b . The area of the rectangle is : S=a*(b-a)=(20-3b)*[b-(20-3b)]=(20-3b)*[b-20+3b]=
(20-3b)*(4b-20)=80b-400-12b²+60b=140b-12b²-400. To obtain maximum, we derive S with respect
to b : S'(b)=140-24b . Now we set the derivative equal zero :
140-24b=0 --> b=5.833 . Let's find a :
a=20-3b=2.5
So, the dimensions of the largest triangle are : 3⅓ X 2½ .

Alon.

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".

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

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M.A in Mathematics & Bs.c in Electronics.

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