PRE-ALGEBRA

PRE-ALGEBRA

Reasoning Strategy: Make a Model (10-8)

How can you use a model to help solve a problem?

Example: A box company makes boxes to hold popcorn for a movie theater. Each box is made from a piece of cardboard that is 8½ in. by 11 in. (the size of a piece of copy paper). If the boxes are made by cutting out a square from the corner and folding the sides up, then what are the dimensions of the box that will hold the most popcorn?

Step 1: Understand the problem.1. Goal: Find the dimensions of the box that will hold the most popcorn = greatest volume2. Given Information: The size of the cardboard is is 8½ in. by 11 in.

Step 2: Make a plan.1. Build boxes using sheets of paper. 2. Cut squares from the corners of increasing sizes starting with whole numbers 1 in., 2 in., 3 in. and so on. 3. Find the dimensions and volumes of each box until the values start to decrease. Then, go by halves.

PRE-ALGEBRA

Reasoning Strategy: Make a Model (10-8)

Size of Cut Length Width Height Volume

1 in. 9 in. 6.5 in. 1 in. 9 6.5 1 = 58.5 in.3

2 in. 7 in. 4.5 in. 2 in. 7 4.5 2 = 63 in.3

3 in. 5 in. 2.5 in. 3 in. 5 2.5 3 = 37.5 in.3

1½ in. 8 in. 5.5 in. 1.5 in. 8 5.5 1.5 = 66 in.3

2½ in. 6 in. 3.5 in. 2.5 in. 6 3.5 2.5 = 52.5 in.3

Step 3: Organize your information into a table or chart.

So, the box that would hold the most popcorn would be 8 in. (length) by 5½ in. (width) by 1½ in. (height) and hold 66 cubic in. (in.3)

PRE-ALGEBRA

A can company rolls rectangular pieces of metal that

measure 8 in. by 10 in. to make the sides of cans. Which height, 8 in.

or 10 in., will make the can with the greater volume?

Build two cans using 8 in.-by-10 in. pieces of paper. You do not need to make the bases, just the sides.

LESSON 10-8

Additional Examples

Reasoning Strategy: Make a Model

PRE-ALGEBRA

(continued)

The volume of the can with height 8 in. is greater.

not to scale

Measure your models to find approximate radii.Radius of 10-in. high can 1.3 in.Radius of 8-in. high can 1.6 in.

Find the volumes.V = r 2h

= (1.3)2(10)

= 16.9

53.1 in.3

V = r 2h

= (1.6)2(8)

= 20.48

64.3 in.3

LESSON 10-8

Additional Examples

Reasoning Strategy: Make a Model

PRE-ALGEBRA

Solve.

1. You cut square corners from a piece of cardboard that has dimensions 32 cm by 40 cm. You then fold the cardboard to create a box with no lid. To the nearest centimeter, what are the dimensions of the box that will have the greatest volume?

20 cm by 28 cm by 6 cm

LESSON 10-8

Reasoning Strategy: Make a Model

Lesson Quiz