Question

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 15 in. long and 7 in. wide, find the dimensions (in inches) of the box that will yield the maximum volume. (Round your answers to two decimal places if necessary.) smallest value in in largest value in

Answer 1

Length l = 15 - 2x = 15 - 2(1.5) = 12.00 in

Breadth b = 7 - 2x = 7 - 2(1.5) = 4.00 in

Height h = x = 1.50 in

Explanation:

The volume of a box can be written as;

V = l×b×h

Where;

Length = l

Breadth = b

Height = h

Let x represent the length of the cube cut out of the four edges.

Using the attached image;

Length l = 15 - 2x

Breadth b = 7 - 2x

Height h = x

Substituting the values to the volume equation;

V = (15-2x)(7-2x)(x)

V = 105x - 30x^2 - 14x^2 + 4x^3

V = 105x - 44x^2 + 4x^3

At Maximum volume, V' = dV/dx = 0

V' = 105 - 88x + 12x^2 = 0

Solving the quadratic equation, we have;

x = 5.83 or x = 1.50

x cannot be 5.83 since 2x > 7 (greater than the breadth of cardboard)

Therefore ;

Length l = 15 - 2x = 15 - 2(1.5) = 12.00 in

Breadth b = 7 - 2x = 7 - 2(1.5) = 4.00 in

Height h = x = 1.50 in