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. 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 cut from each corner be x cm. The length of the resulting box will be (40-2x) cm and the width will be (30-2x) cm. The height of the box will be x cm. Therefore, the equation that models this situation is:
Volume = (40-2x)(30-2x)(x) = 1872 cm^3
b) To solve this equation, expand the terms in the equation and simplify to get:
Volume = 4x^3 - 140x^2 + 1200x = 1872
4x^3 - 140x^2 + 1200x - 1872 = 0
One technique to solve this cubic equation is to use trial and error to find the solutions. Another technique is to use factorization or synthetic division.
c) Not all solutions to the equation may be possible lengths of the square that Nathan is going to cut out of the rectangle because the dimensions of the resulting box have to be positive. Any negative values for x would result in negative dimensions, which is not possible in this case.
d) By solving the equation 4x^3 - 140x^2 + 1200x - 1872 = 0, the solution for x is x = 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.