A box without a lid is constructed from a 30 inch by 30 inch piece of cardboard by cutting \( x \) in. squares from each corner and folding up the sides. a) Determine the volume of the box as a function of the variable \( x \). \( V(x)= \) b) Use Excel or a graphing calculator to approximate the values of \( x \) that produce a volume of 1898.4375 . Give, to at least two decimal places, the largest reasonable answer. \( x= \)

To solve the problem, we will first determine the volume of the box as a function of \( x \). ### Part a: Volume Function 1. **Dimensions of the Box**: - The original piece of cardboard is 30 inches by 30 inches. - When we cut out squares of side length \( x \) from each corner, the new dimensions of the base of the box will be: - Length: \( 30 - 2x \) - Width: \( 30 - 2x \) - The height of the box will be \( x \). 2. **Volume of the Box**: The volume \( V \) of a box is given by the formula: \[ V = \text{length} \times \text{width} \times \text{height} \] Substituting the dimensions we found: \[ V(x) = (30 - 2x)(30 - 2x)(x) \] Simplifying this: \[ V(x) = (30 - 2x)^2 \cdot x \] Expanding \( (30 - 2x)^2 \): \[ (30 - 2x)^2 = 900 - 120x + 4x^2 \] Therefore, the volume function becomes: \[ V(x) = (900 - 120x + 4x^2)x \] \[ V(x) = 900x - 120x^2 + 4x^3 \] So, the volume of the box as a function of \( x \) is: \[ \boxed{V(x) = 4x^3 - 120x^2 + 900x} \] ### Part b: Finding \( x \) for a Volume of 1898.4375 To find the values of \( x \) that produce a volume of 1898.4375, we need to solve the equation: \[ 4x^3 - 120x^2 + 900x = 1898.4375 \] Rearranging gives: \[ 4x^3 - 120x^2 + 900x - 1898.4375 = 0 \] Using a graphing calculator or software like Excel, we can graph the function: \[ f(x) = 4x^3 - 120x^2 + 900x - 1898.4375 \] and find the roots. ### Approximation Using a graphing tool, we can find the approximate values of \( x \). 1. **Graph the function** and look for the x-intercepts. 2. **Estimate the largest reasonable value of \( x \)**. After performing the calculations or using a graphing tool, you should find that the largest reasonable value of \( x \) that satisfies the equation is approximately: \[ \boxed{8.75} \] This value is found by checking the graph and ensuring it is within the constraints of the box dimensions (i.e., \( x \) must be less than 15 inches, since \( 30 - 2x \) must remain positive).