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 side x cm in length from each corner, as shown in the picture below. 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 side of the square cut out from each corner be x cm. The dimensions of the resulting box will be (40-2x) cm by (30-2x) cm by x cm. Therefore, the volume of the box can be calculated as V = x(40-2x)(30-2x).
b) To solve the equation V = x(40-2x)(30-2x) = 1872 cm^3, we can expand the equation to get 4x^3 - 140x^2 + 1200x - 1872 = 0. This is a cubic equation, which can be solved using techniques such as factoring, the rational root theorem, or a graphing calculator. In this case, using a graphing calculator would be the most efficient method.
c) Not all solutions to the cubic equation will be valid lengths for the side of the square. We must consider that the length of the side of the square cannot exceed half the length of the original rectangle (x ≤ 15 cm). Additionally, the length of the resulting box cannot be negative, so only the positive real solutions to the equation will be valid.
d) The length of a side of the square that Nathan is going to cut from the corners of the rectangle is x = 6 cm.