Question

Ramesh, the owner of a sweet selling shop, purchased some rectangular cardboard sheets of dimension 25 cm by 40 cm to make container packets without tops. Let x cm be the length of the side of the square to be cut out from each corner to give that sheet the shape of the container by folding up the flaps.

Based on the above information, answer the following questions:

1. Express the volume (V) of each container as function of x only.  (1)
2. Find `(dV)/dx`.  (1)
3.    
   1. For what value of x, the volume of each container is maximum?  (2)
                               OR
   2. Check whether V has a point of inflection at x = `65/6` or not?  (2)

Answer 1

i. V = (40 − 2x)(25 − 2x)x cm³

ii. `(dV)/dx = 4(3x - 50)(x - 5)`

iii. (a) For extreme values, `(dV)/dx = 4(3x - 50)(x - 5) = 0`

⇒ `x = 50/3  "or"  x = 5`

`(d^2V)/dx^2 = 24x - 260`

`therefore (d^2V)/dx^2` at x = 5 is −140 < 0

∴ V is max when x = 5.

OR

(b) For extreme values, `(dV)/dx = 4(3x^2 - 65x + 250)`

`(d^2V)/dx^2 = 4(6x - 65)`

`(dV)/dx  "at"  x = 65/6  "exists and"  (d^2V)/dx^2  "at"  x = 65/6` is 0.

`(d^2V)/dx^2  "at"  x = (65/6)^- "is negative and"  (d^2V)/dx^2   "at"  x = (65/6)^+`is positive.

`therefore x = 65/6` is a point of inflection.