Applications of Cubic Functions
Volume of a Open Box.
Suppose you are trying to make an open-top box out of a piece of cardboard that is 20 inches by 16 inches. You are to cut the same size square from each corner. Write a function to represent the volume of this box.
20
16
16 -
2x
20 - 2xx
x
x
x
x
x
xx
V=lwh
20 - 2x
20 - 2x16 - 2x
?x
))(216)(220( xxxV
Formula for the Volume of a Box
))(2)(2( xxWxLV The final answer for the volume will
ALWAYS have the term :
34x
20
-2x
16 -2x
Write the formula for the volume of our box:
))(216)(220( xxxV
xxxV )320724( 2
Step 1: Multiply the two binomials together
Step 2: Multiply by x
320
-32x
-40x
4x2 xxxV 320724 23
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 < 8 (Half of the length of the smallest side)
• SO, Xmin = 0 and Xmax = 8; ZOOM 0
• Do you want x or y?
• Y!!!
• 420 cubic inches
What size square should be cut from each corner to realize the
maximum volume?
• What do you want now?
• X!!
• 2.9 inches
What size square should you cut from each corner to realize a volume of 300 cubic inches?
• What do you know: x or y?
• Y!! Let y = 300; find the intersection
• 1.3 inches or 5 inches
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
• 384 Cubic inches