## By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may b

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

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2021-10-09T01:10:18+00:00
2021-10-09T01:10:18+00:00 1 Answer
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## Answers ( )

Answer:

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

Step-by-step 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