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Maximizing Volume of a Box: Cubic Regression and Application Problems

This guide explores how to optimize the volume of an open box created from a 10x12 inch piece of cardboard by cutting corners. We formulate the volume equation in terms of 'x' and determine the greatest achievable volume using cubic regression. By labeling side lengths, writing in factored form, and utilizing technology to find the maximum volume, we identify that cutting 1.811 inches from each corner yields a maximum volume of approximately 96.771 cubic inches. The analysis also covers why volume cannot increase indefinitely and details the resulting box dimensions.

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Maximizing Volume of a Box: Cubic Regression and Application Problems

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  1. Notes Day 6.5 Choose the Best Regression Equation Volume of a Box cubic Application Problems

  2. Complete a regression equation to find the best model. Go to “catalog” on the calculator and turn “diagnostics on.” Quadratic: .9304 R2= Y = -9.326x2 + 109.571x – 20.288 Cubic: 1 R2= Y = 5x3 – 9x2– (2.4• 10-11)x + 8 1 but not really quartic R2= Quartic: Y = (5.57• 10-12)x4 + 5x3– 9x2 + 8 Cubic Regression is best

  3. An open box is to be made from a 10-in. by 12-in. piece of cardboard by cutting x-inch squares in each corner and then folding up the sides. Write a function giving the volume of the box in terms of x. Approximate the value of x that produces the greatest volume. A. Label the side lengths in terms of x 12 – 2x Write an equation in factored form for the volume as if the box were closed. x x V(x)=(12 – 2x)(10 – 2x)(x) C. Find the roots and plot on the graph. 10 – 2x X = {6,5,0} Write the volume equation in standard form and plot the end behavior on the graph. V(x)=(120 – 44x + 4x2)(x) V(x)=4x3 – 44x2 + 120x 90 E. Find the relative min/max on the calculator. Plot. (1.811 , 96.771) Min(5.523 , - 5.511) F. Explain what this ordered pair represents? Cuts of 1.811 inches will maximize volume to 96.771 cubic inches 5 Why isn’t volume greatest as x approaches infinity? Some dimensions would be negative. G. What are the dimensions of the box? 8.378 in. by 6.378 in. by 1.811 in.

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