Download 2.16-17.10 Maximum Volume of a Box At the end of the day students

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