Consider a cuboid of sides $2x$,$4x$,and $5x$ and a closed hemisphere of radius $r$. If the sum of their surface areas is a constant $k$,then the ratio $x:r$,for which the sum of their volumes is maximum,is

$A$ wire of length $22 \; m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then,the length of the side of the equilateral triangle,so that the combined area of the square and the equilateral triangle is minimum,is

Given $f(x) = -\frac{x^3}{3} + x^2 \sin(1.5a) - x \sin(a) \sin(2a) - 5 \sin^{-1}(a^2 - 8a + 17)$,then:

At what value of $x$ does the function $f(x) = x^x$ $(x > 0)$ attain its minimum value?

If the functions $f(x) = \frac{x^3}{3} + 2bx + \frac{ax^2}{2}$ and $g(x) = \frac{x^3}{3} + ax + bx^2$,where $a \neq 2b$,have a common extreme point,then $a + 2b + 7$ is equal to

If the lengths of three sides of a trapezium other than the base are equal to $10 \ cm$,find the maximum area of the trapezium.