Page:Calculus Made Easy.pdf/130

 (3) A line of length $$p$$ is to be cut up into $$4$$ parts and put together as a rectangle. Show that the area of the rectangle will be a maximum if each of its sides is equal to $$\tfrac{1}{4}p$$.

(4) A piece of string $$30$$ inches long has its two ends joined together and is stretched by $$3$$ pegs so as to form a triangle. What is the largest triangular area that can be enclosed by the string?

(5) Plot the curve corresponding to the equation

also find $$\dfrac{dy}{dx}$$, and deduce the value of $$x$$ that will make $$y$$ a minimum; and find that minimum value of $$y$$.

(6) If $$y=x^5-5x$$, find what values of $$x$$ will make $$y$$ a maximum or a minimum.

(7) What is the smallest square that can be inscribed in a given square?

(8) Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder (a) whose volume is a maximum; (b) whose lateral area is a maximum; (c) whose total area is a maximum.

(9) Inscribe in a sphere, a cylinder (a) whose volume is a maximum; (b) whose lateral area is a maximum; (c) whose total area is a maximum.