A rectangular sheet of fixed perimeter with sides having their lengths in the ratio `8: 15` is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are 24 (b) 32 (c) 45 (d) 60

Options

A. 24,45
B. 32,65
C. 24,60
D. 32,60

Answer

A

Explanation

Let the sides of the rectangular sheet be 15a and 8a units and the length of the side of each square to be cut from each corner of the sheet be x units .Then the dimension of the box are
Length =15a-2x ,Breadth = 8a -2x,Depth =x Clearly Length `gt 0` Breadth `gt 0` and Depthe `gt 0` . Therefore
`0 lt x lt4a`

Let V be the volume of the box. Then,
`V=(15a-2x)(8a-2x)x=120a^x-46 ax^2+4x^3`
`therefore (dV)/(dx)=120a^2-92ax+12x^2 and (d^2V)/(dx^2)=-92ax+12x^2 and (d^2V)/(ax^2)=-92 a+ 24 x `
The critical numbers of V are given by `(dV)/(dx)=0`
`therefore (dV)/(dx)=120a^2-92 x +12x^2 and (d^2V)/(dx^2)=-92a+24x`
The critical numbers of V are given by `(dV)/(dx)=0`
` therefore (dV)/(dx)=0`
`rArr 120a^2 -92ax+12x^2=0`
`rArr 30a^2-23ax+3x^2 =0`
`rArr (6a-x)(5a-3x)=0`
`rArr 5a-3x=0`
`rArr x =(5a)/(3) " "[ because 0 lt x lt 4a therefore 6a- x in 0 ]`
When `x(5a)/(d^2V)/(dx^2)=-92 a+ 40a=-52a lt 0 `
It is given that the total area of all squares cut form each coner of the sheet is 100 sq. units .
Hence , the demensions of the sheet are `8a=24 and 15 a = 45 ` units .