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Calculus/calculus max/min problems
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Question
There are 2 problems that i can not seem to figure out.
1. Find the dimensions of the rectangle of maximum area that can be inscribed in a semicircle or radius a, if two vertices lie on the diameter.
2. A commercial cattle rance currently allows 20 steers per acre of grazing land; on the average its steers weigh 2000lb at market. Estimates by the Agriculture Department indicate that the average market weight per steer will be reduced by 50lb for each additional steer added per acre of grazing land. How many steers per acre should be allowed in order for the ranch to get the largest possible total market weight for its cattle?
Thank you so much.
1.
)
As Its drawn in the image I attached, we conclude that :
y=(a^2-x^2)^(1/2). The area of the rectangle is S=2x*y whic is
2x*(a^2-x^2)^(1/2). We need to maximize this function, so we derive & set the derivative equal zero:
S'(x)=2(a^2-x^2)^(1/2)-2x*2x/2(a^2-x^2)^(1/2).
S'(x)=0 -> 2(a^2-x^2)^(1/2)=4x^2/2(a^2-x^2)^(1/2).
(a^2-x^2)=x^2 -> 2x^2=a^2 -> x=a/2.
As for 2, I didn't understand the problem well .. please try to explain it again. Thank you
Alon.
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Answers by Expert:
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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