A truck rental company rents a 12 ft by 8 ft by 6 ft truck for \( \$ 69.24 \) per day plus mileage. A customer prefers to rent a less expensive smaller truck whose dimensions are \( x \mathrm{ft} \) smaller on each side. If the volume of the smaller truck is \( 240 \mathrm{ft}^{3} \), determine the dimensions of the smaller truck.

To find the dimensions of the smaller truck, we start by calculating the volume of the larger truck. The volume \( V \) of a rectangular prism is given by \( V = \text{length} \times \text{width} \times \text{height} \). For the larger truck: \[ V = 12 \, \text{ft} \times 8 \, \text{ft} \times 6 \, \text{ft} = 576 \, \text{ft}^3 \] Now, if the dimensions of the smaller truck are \( (12 - x) \) ft, \( (8 - x) \) ft, and \( (6 - x) \) ft, we can set up the equation based on the volume of the smaller truck provided as \( 240 \, \text{ft}^3 \): \[ (12 - x)(8 - x)(6 - x) = 240 \] Expanding the left side can be a bit tricky, but let's go step by step. Firstly, we can find the factor \( (12 - x)(8 - x) \): \[ (12 - x)(8 - x) = 96 - 20x + x^2 \] Next, we multiply this by \( (6 - x) \): \[ (96 - 20x + x^2)(6 - x) = 576 - 96x - 120x + 20x^2 + 6x^2 - x^3 \] \[ = 576 - 216x + 26x^2 - x^3 \] Now we will set this equal to \( 240 \): \[ 576 - 216x + 26x^2 - x^3 = 240 \] Rearranging gives us: \[ -x^3 + 26x^2 - 216x + 336 = 0 \] Multiplying new by -1 to simplify factors out: \[ x^3 - 26x^2 + 216x - 336 = 0 \] We can use synthetic division or numerical methods to find the roots. However, through trial and error or calculator assistance, we find one root to be \( x = 6 \): Substituting back into the equation, The smaller truck dimensions are thus: \[ (12-6) \text{ ft}, (8-6) \text{ ft}, and (6-6) \text{ ft} \] which gives us dimensions of \(6 \text{ ft}, 2 \text{ ft}, and 0 \text{ ft}\), which is incorrect since dimensions cannot be zero. This indicates \(x\) found is too large: retry small values of \(x\), where feasible \(1\) or \(2\). For instance, trying x = 2: \[ (12-2)(8-2)(6-2) = 10 * 6 * 4 = 240 \] Thus, the real dimensions of the smaller truck are: \(10 \text{ ft}, 6 \text{ ft}, 4 \text{ ft}.\)