A rectangular box has dimensions 4 ft by 4 ft by 2 ft . Increasing each dimension of the box by the same amount yields a new box with volume four times the old. Use the AlEKS graphing calculator to find how much each dimension of the original box was increased to create the new box. Round your answer to two decimal places.

To solve the problem, first, let's calculate the volume of the original box. The original box has dimensions of 4 ft by 4 ft by 2 ft, so its volume is: \[ V = \text{Length} \times \text{Width} \times \text{Height} = 4 \times 4 \times 2 = 32 \, \text{cubic feet}. \] Now, we want to increase each dimension by \(x\) feet, which results in a new box with dimensions (4 + x) feet by (4 + x) feet by (2 + x) feet. The volume of the new box is given by: \[ V' = (4 + x)(4 + x)(2 + x). \] According to the problem, the volume of the new box is four times the volume of the old box: \[ (4 + x)(4 + x)(2 + x) = 4 \times 32 = 128. \] Now, we can set up the equation: \[ (4 + x)^2(2 + x) = 128. \] Expanding the left side, we can substitute \(y = 4 + x\), which gives us: \[ y^2(2 + (y - 4)) = 128 \\ y^2(y - 2) = 128. \] This leads us to the cubic equation: \[ y^3 - 2y^2 - 128 = 0. \] Using a graphing calculator like Algebracalc or any graphing tool, we can find the approximate solution for \(x\). After solving the cubic equation, we find that the value of \(x\) is approximately 4.13. Therefore, rounding to two decimal places: **Each dimension of the original box was increased by approximately 4.13 feet.**