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2.16-17.10 Maximum Volume of a Box At the end of the day students will…write cubic functions in factored form, multiply factors to get standard form of polynomials Standards Taught… identifying patterns Anticipation of next steps… Identifying polynomials of varying degrees Dividing polynomials Synthetic division Warm-Up…find the equation for this table. x y -3 -7 Now check with your calculator to see if it -2 -6 generates this table then graph -1 -5 0 2 Notice the pivot point(point of inflection) 1 21 this is the point from which we want to write the function 2 58 Form y A R( x w)3 3 15 10000 Step by Step Instruction… y A R( x w)3 Cubics can be written in different forms y Ax3 Bx 2 Cx D y x( x m)( x n) We need to be able to move between the forms just like with quadratics Students do maximum volume of a box to investigate cubics and their different forms Independent Practice…students work in groups to complete worksheet 1 FST Name:_______________Per:___ Investigation: Maximum Volume of a Box Guiding Question: What is the maximum volume of an open-topped box made from a single sheet of paper? Part 1 A box with an open top can be made from an 8.5 by 11 inch piece of paper by cutting congruent squares from the four corners and folding up the sides as shown below. x x 8.5” 11” 1. Cut out square corners when x = 0.5 inches, fold the sides up and then calculate the volume of the box. x x Volume of the box = ___________ in3 2 Part 2 1. Unfold the box and cut another section out so that each corner has a 1 inch square cut out of it. Again, calculate the volume. Do this for each 0.5 inch increment and record the data in the table below. Height (in) 0.5 1 1.5 2 2.5 3 3.5 4 2. Width (in) Length (in) Volume (in3) Plot the Volume vs. Height on the graph provided below and connect the points with a smooth curve. Volume of the Box 80 70 Volume (in3) 60 50 40 30 20 10 2 3 4 1 Height (in) 3. What happens to the volume of the box as the height gets larger? _____________________ Why do you think this is the case? 4. According to your data, what are the dimensions of the box with the maximum volume? ____ in. x ____ in. x ____ in. 5. What is the maximum volume of the box? ____________ in3 3 6. Find the equation for the volume of the box in factored form. Equation:_________________ 7. Find the equation for the volume of the box in standard form. (Multiply the terms from factored form) Equation:_______________ 8. Enter the height and volume data into your calculator and graph the data points. Enter your equations from problems 6 and 7 and check to see if they fit the data. Sketch your graph in the space below. Show your graph and equations to your teacher. Teacher’s signature:____________________ 9. Find the equation for the volume of the box below in factored form. x x 15” 25” 10. Change the equation from problem 9 above to standard form. 4