Questions and Solutions
Question 1:
Find the maximum area of an isosceles triangle inscribed in the ellipse \( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \) with its vertex at one end of the major axis.
Question 2:
A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is \( 2\,\text{m} \) and volume is \( 8\,\text{m}^3 \). If the building of tank costs Rs 70 per square meter for the base and Rs 45 per square meter for the sides, what is the cost of the least expensive tank?
Question 3:
The sum of the perimeter of a circle and a square is \( k \), where \( k \) is some constant. Prove that the sum of their areas is least when the side of the square is double the radius of the circle.
Question 4:
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is \( 10\,\text{m} \). Find the dimensions of the window to admit maximum light through the whole opening.
Question 5:
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius \( r \) is \( \dfrac{4r}{3} \).
Question 6:
Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius \( R \) is \( \dfrac{2R}{\sqrt{3}} \). Also, find the maximum volume.
Question 7:
Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height \( h \) and semi-vertical angle \( a \) is one-third that of the cone and the greatest volume of the cylinder is \( \dfrac{4}{27} \pi h^2 \tan^2 a \).