Question

The ratio of adjacent sides of a rectangular cardboard is 1:2. A square of area 4 cm2is cut out from each corner of the cardboard. The remaining cardboard is folded to form an open cuboid box of height 2 cm. If the total outer surface area of the open box is 82 cm2, what is the volume of the box (in cm3)?​

Answer 1

The volume of the open box is 65 $cm^{3}$ .

Step-by-step explanation:

Given:

Adjacent sides of a rectangular cardboard have ratio of 1:2.

Each corner of the cardboard is cut  a square of area 4 cm2.

The remaining cardboard of height 2cm is folded to form an open cuboid box.

Total outer surface area of the open box is 82 cm^2.

To Find:

The volume of the open box.

Formula Used:

Volume of a cuboid V= m × n × h

Total Surface Area of a  open Cuboid  S  = m x n+2 m x h +2 n x h

Here,        m= length  ,   n= width ,                    h = height

Solution:

As given-  ratio of adjacent sides of a rectangular cardboard is 1:2.

Let  length of rectangular cardboard is 2p.  Width of cardboard  will be p.

As given - each corner of the cardboard is cut  a square of area 4 cm2.

Therefore,  length of open cuboid box  m = 2p-8

Width of open cuboid box n = p-8

Height of open cuboid box h = 2 cm

Total outer surface area of the open box S= 82

Total outer surface area of the open box S = m x n+2 m x h +2 n x h

= (2 p-8) (p-8)+2(2 p-8)2+2(p-8)2

=2 p^2-8 p-16 p+64+8 p-32+4 p-32

82  = 2 p^2-12 p

p^2-6 p-41=0   ( Solving quadratic  equation by formula)

$p = \frac{-6(+/-) \sqrt{(-6)^{2} - 4.1.(-6) } }{2.1}$

$p = \frac{-6(+/-) \sqrt{200} }{2}$

p= 3 (+/-) 5$\sqrt{2}$

p=  10.05 cm  and (- 4.07 cm)

Considering positive value  of p.

p= 10.05 Cm

Volume of the  open box V = m × n × h

=(p-8)x (p-8) x2

=(2.10.5-8)x (10.05-8)x 2

= (13)x(2.5)x 2 $cm^{3}$

 = 65 $cm^{3}$

Thus, the volume of the open box is 65 $cm^{3}$ .