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