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Multivariable calculus - triangle

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Consider a triangle in the plane, with angles , a, b , c.
Assume that the radius of its circle is equal to 1.

1) by decomposing the triangle into six right triangles having the incenter as a common vertex, express the area A of the triangle in term of a, b , c ( the answer should be a symmetric expression). Then use the result to show that A can be expressed as a function of the two variables a and b by the formula

A = cot(a/2) + cot(b/2) + tan((a+b)/2)

b) what is the set of possible values for a and b ? Find all the critical points of the function A in this region.

c) By computing the values of A at the critical points and its behavior on the boundary of the region where it is defined, find the maximum and the minimum of A (could you please justify your answer).
Describe the shapes of the triangle corresponding to these two situations.

d) use the second derivative test to confirm the nature of the critical points you found in b)

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Solution Summary

This shows how to create a symmetric expression for a situation, then find the set of values for variables and critical points.

Solution provided by:
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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