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.
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
Bonus Knowledge
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: \(10 \text{ ft}, 6 \text{ ft}, 4 \text{ ft}.\)
preguntas relacionadas
