1) A designer is creating various open-top boxes by cutting four equally-sized squares from the corners of a 21-cm by 32-cm sheet of cardboard, and then folding up and securing the resulting \'flaps\' to be the sides of the box. a) What\'s the maximum volume of the box? b) What\'s the side length of the square cutout that achieves this maximum volume in part a)? 2)The cost of manufacturing a semi-circular window is the cost of the glass pane together with the cost of the framing material. When ordering this window, a customer specifies the length of the base of the window, shown as l Solution let the size of the square to cut is = x then the volume is = (21-2x)*(32-2x)*x so we have to maximize (21-2x)*(32-2x)*x by differentiating and putting =0 we get 4.14 so maximum = 1250 a) 1250 b) 4.14 2) let rate for glass is g$/sq and rate for framing material = f$/unit length I is the base so radius is I/2 so cost is = pi*I/2 * f + pi*(I/2)^2 * g.

1. 1) A designer is creating various open-top boxes by cutting four equally-sized squares from the
corners of a 21-cm by 32-cm sheet of cardboard, and then folding up and securing the resulting
'flaps' to be the sides of the box.
a) What's the maximum volume of the box?
b) What's the side length of the square cutout that achieves this maximum volume in part a)?
2)The cost of manufacturing a semi-circular window is the cost of the glass pane together with
the cost of the framing material. When ordering this window, a customer specifies the length of
the base of the window, shown as l
Solution
let the size of the square to cut is = x
then the volume is = (21-2x)*(32-2x)*x
so we have to maximize (21-2x)*(32-2x)*x
by differentiating and putting =0 we get 4.14
so maximum = 1250
a) 1250
b) 4.14
2)
let rate for glass is g$/sq and rate for framing material = f$/unit length
I is the base so radius is I/2
so cost is = pi*I/2 * f + pi*(I/2)^2 * g