2. Applications of Cubic Functions

3. Volume of a Open Box. Suppose you are trying to make an open-top box out of a piece of cardboard that is 12 inches by 9 inches. You are to cut the same size square from each corner. Write a function to represent the volume of this box. 12 9 9 - 2x 12 - 2x x x x x x x x x

4. V=lwh 12 - 2x 9 - 2x x

5. Formula for the Volume of a Box The final answer for the volume will ALWAYS have the term :

6. Write the formula for the volume of our box: Step 1: Multiply the two binomials together Step 2: Multiply by x

7. What is the maximum volume? What is the possible domain for this box? What is the greatest possible value that we can cut out for x? 0 < X < 4.5 (Half of the length of the smallest side) SO, Xmin = 0 and Xmax = 4.5; ZOOM 0 Do you want x or y? Y!!! 81.9 cubic inches

8. What size square should be cut from each corner to realize the max volume? What do you want now? 1.70 inches What are the dimensions of the box with the maximum volume? 12 - 2x 12 - 2(1.7) 12 - 3.4 8.6 9 - 2x 9 - 2(1.7) 9 - 3.4 5.6 8.6 x 5.6 x 1.7 x 1.7 X!!

9. What size square should you cut from each corner to realize a volume of 50 cubic inches? What do you know: x or y? Y!! Let y = 50; find intersection.59 inches or 3.1 inches

10. What is the volume if a square with side 2 inches is cut from each corner? What do you know; x or y? X!!! Go to table; let x = 2 80 Cubic inches

11. American Indians For 1890 through 1990, the American Indian, Eskimo, and Aleut population P (in thousands) can be modeled by the function: where t is the number of years since 1890. 1)What is t? What is that in calc? 2)What is P? What is that in calc? 3)In what year did the population reach 722,000? population; y years; x Year 75 = 1965

12. The volume of a rectangular swimming pool is modeled by the polynomial : If width of the pool is given by the polynomial 4t – 3, what polynomials express the length and depth? Answers: (t – 7)(t + 1) Use Synthetic or Long Division

13. The data compares X, the areas in square feet of ten houses, with Y, the electricity consumptions in kilowatt-hours per month. X 1290165017381829205220932399247626532786 Y1007140817241748162916831704190419562215 1. Find a regression model to fit this data. 2. Predict the kilowatt-hours per month for a house with 3500 square feet. Does this amount make sense?