A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8:15 is converted into an open rectangular box by folding after removing squares of equal areas 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:

THIS QUESTION HAS MULTIPLE CORRECT OPTIONS
A. 24
B. 32
C. 45
D. 60

We are given that the perimeter of a rectangular sheet is fixed, and its sides are in the ratio 8:15. After removing equal-area squares from all four corners and folding the sheet to form an open rectangular box, the total area of the removed squares is 100, and the resulting box has a maximum volume. We need to find the length of the sides of the rectangular sheet.

Let the lengths of the sides of the rectangular sheet be:

Length = 8x
Width = 15x
The perimeter of the rectangle is given by:

Perimeter = 2 * (Length + Width) Perimeter = 2 * (8x + 15x) = 2 * 23x = 46x

We are not explicitly given the value of the perimeter, but we know the total area of removed squares is 100. This means that the area of each square is equal, and the sum of the areas of the four squares is 100.

Let the side length of each square be "y". Then, the total area of the squares removed is:

4 * y^2 = 100 y^2 = 25 y = 5

Now, the new length and width of the rectangular box after removing the squares will be:

New length = 8x - 2y = 8x - 2 * 5 = 8x - 10
New width = 15x - 2y = 15x - 2 * 5 = 15x - 10
Height of the box = y = 5
The volume of the box is given by the product of its length, width, and height:

Volume = (8x - 10) * (15x - 10) * 5

To find the value of x that maximizes the volume, we first expand the expression for the volume:

Volume = 5 * (8x - 10) * (15x - 10) Volume = 5 * (120x^2 - 80x - 150x + 100) Volume = 5 * (120x^2 - 230x + 100) Volume = 600x^2 - 1150x + 500

To maximize the volume, we differentiate the volume with respect to x and set the derivative equal to zero:

dV/dx = 1200x - 1150

Set the derivative equal to zero:

1200x - 1150 = 0 1200x = 1150 x = 1150 / 1200 x = 23 / 24

Now substitute this value of x into the expressions for the length and width of the rectangular sheet:

Length = 8x = 8 * (23 / 24) = 184 / 24 = 7.67 Width = 15x = 15 * (23 / 24) = 345 / 24 = 14.38

Thus, the sides of the rectangular sheet are approximately:

Length = 7.67 units Width = 14.38 units

The correct answer will depend on further clarification or assumptions about the exact numerical values of the sheet.