I need help maximizing a triangle
- Thread starter kitkalin
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We need to see your work - where are you getting stuck?
Have you written an equation of the Area? Can you use the Pythagorean theorem? How do you generally find a max value?
pka
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Maximize \(\displaystyle \frac{1}{2}\sqrt{100x^2-x^4}\); realize that \(\displaystyle x>0\).
An optimizing problem in differential calculus always involves setting up an expression that equals the variable to be optimized. If the resulting equation has only one independent variable, you simply take the first derivative of the variable to be optimized with respect to the independent variable, set the derivative to zero, and solve for the independent variable algebraically. (There are some complications with boundary conditions and inflection points, but the basic idea is to set the first derivative to zero and solve.) So it is scary when you have the equation staring at you and say you do not know what to do next.
You can eliminate \(\displaystyle x = - \sqrt{100 - y^2}\) from consideration because length is not negative.
pka found y in terms of x rather than y in terms of x. It makes no difference which way you go, but here is your way.
\(\displaystyle A = \dfrac{1}{2} * xy = \dfrac{1}{2} * \sqrt{100 - y^2} * y = \dfrac{1}{2} * \sqrt{100 - y^2} * \sqrt{y^2} = \dfrac{1}{2} \sqrt{y^2(100 - y^2)} = \dfrac{1}{2} * \sqrt{100y^2 - y^4}.\)
Now compute \(\displaystyle \dfrac{dA}{dy}\) and proceed.