Question

An open box is to be made from a rectangular piece of cardboard that measures 6 in. by 3 in., by cutting out squares of the same size from each corner and bending up the sides. Is it possible to cut the squares so that the volume of the box is 40 in.3? Find all real solutions of this equation to answer the question. (6 – 2x)(3 – 2x)x = 40

Answer 1

To find out if it's possible to cut squares from a cardboard to form an open box with volume 40 in³, solve the quadratic equation (6 – 2x)(3 – 2x)x = 40, which gives x = 2.5 as the real root. Thus, it is possible to cut squares of size 2.5 in from each corner to form the open box with a volume of 40 in³.

Step-by-step explanation:

To determine whether it is possible to cut squares from each corner of the rectangular cardboard to form an open box with a volume of 40 in³, we need to solve the equation (6 – 2x)(3 – 2x)x = 40.

1. Expand the equation and rearrange it to a quadratic equation: 4x^3 - 18x^2 + 18x - 40 = 0.
2. Factor the quadratic equation by finding the roots. In this case, the only real root is x = 2.5.

Since the value of x is positive, it is possible to cut squares of size 2.5 in from each corner to form an open box with a volume of 40 in³.

Answer 2

No. The only real solution is x = 4. It is not possible to cut squares of this size.