Q1. A square is to be inscribed in the ellipse
\(\frac{x^2}{16}\)+\(\frac{y^2}{25}\) = 1
What must be the dimensions of the square be to maximize its area? What is the maximum area of the square?
Ans: Side of square = \(\frac{2}{\sqrt{\frac{1}{16}+{\frac{1}{25}}\) = 6.25
Q2. What point on the line with equation, y=5+3x , is closest to the origin O?
Ans : a = x + y = x +(5 + 3x) 2a\(\frac{da}{dx}\) = 2x + 6(5 +3x)
\(\frac{da}{dx\) = 0 x = –\(\frac{3}{2}\) , y = \(\frac{1}{2}\)
Q3.Determine two positive numbers such their product is equal to 20 and their sum is minimum?
Ans : xy = 20 s = x + y s = x + \(\frac{20}{y}\) –\(\frac{ds}{dx}\) = 1 —\(\frac{20}{x^2}\)
\(\frac{ds}{dx}\) = 0 x2 = 20 x = \(\sqrt{20}\) = 2\(\sqrt{5}\) y = 2\(\sqrt{5}\)
Q4. Find the maximum area of a rectangle having base on the x-axis and upper vertices on the parabola.
Ans : a =
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