Final answer:
The dimensions of the original 4 ft by 4 ft by 3 ft rectangular box need to be increased by 2 feet each to create a new box with a volume seven times greater, resulting in a new volume of 336 cubic feet.
Step-by-step explanation:
We need to determine by how much each dimension of a rectangular box must increase so that the volume of the new box is seven times that of the original. The volume of the original box is 4 ft by 4 ft by 3 ft, which equals 48 cubic feet. Since the new volume is seven times the old volume, the new volume will be 336 cubic feet.
Let's call the increase in each dimension 'x'. So, the new dimensions will be (4 + x) ft, (4 + x) ft, and (3 + x) ft. To find the new volume, we multiply these dimensions together:
(4 + x)(4 + x)(3 + x) = 336
Then expand and solve the equation:
16(3+ x) + 8x(3 + x) + x^2(3 + x) = 336
48 + 16x + 12x + 4x^2 + 3x^2 + x^3 = 336
x^3 + 7x^2 + 28x - 288 = 0
By using the rational root theorem, synthetic division, or numerical methods such as a graphing tool, we can determine the positive value of 'x' that solves this cubic equation.
Once solved, we find that 'x' is equal to 2 ft. Therefore, each dimension of the original box was increased by 2 feet to create the new box with a volume seven times the old.