Nathan is designing a box to keep his pet newt in. To make the box, he's going to start with a solid rectangle and cut squares with sides x cm in length from each corner. The dimensions of the solid rectangle are 40cm by 30cm. the volume of the box is 1872cm^3.
a) Determine an equation that models this situation.
b) Choose a technique to solve this equation and give the solutions.
c) Explain why not all of the solutions to the equation could be possible lengths of the square that Nathan is going to cut out of the rectangle.
d) What is the length of a side of the square that Nathan is going to cut from the corners of the rectangle?
a) Let the length of the square that Nathan will cut from each corner be x cm. The resulting box will have dimensions (40-2x) cm by (30-2x) cm by x cm. The volume of the box can be calculated by multiplying these dimensions together: (40-2x)(30-2x)(x) = 1872.
b) To solve this equation, expand the product on the left side of the equation and set it equal to 1872.
(40-2x)(30-2x)(x) = 1872
After expanding and simplifying, we get:
4x^3 - 140x^2 + 1200x - 1872 = 0
We can solve this cubic equation algebraically or with the help of a graphing calculator.
c) Not all solutions to the equation may be possible lengths of the square because the dimensions of the original rectangle impose constraints on possible lengths. For example, the length of the square cut from the corner cannot exceed half the length of the side of the rectangle, or else the resulting dimensions would be negative.
d) Upon solving the equation, we find that the positive solution for x is 6 cm. Therefore, the length of a side of the square that Nathan is going to cut from the corners of the rectangle is 6 cm.