Math Problem Statement

Robert is filling a box that is a rectangular prism with a height of 1 1/3 meters, a length of 1 meter, and width of 2/3 meters. How many 1/3 meter cubes will Robert need to fill the box?

Solution

To determine how many 13\frac{1}{3}-meter cubes Robert will need to fill the box, follow these steps:

Step 1: Calculate the volume of the rectangular prism

The volume of a rectangular prism is calculated as: Volume=height×length×width.\text{Volume} = \text{height} \times \text{length} \times \text{width}. Substituting the given dimensions: Volume=43×1×23=43×23=89cubic meters.\text{Volume} = \frac{4}{3} \times 1 \times \frac{2}{3} = \frac{4}{3} \times \frac{2}{3} = \frac{8}{9} \, \text{cubic meters}.

Step 2: Calculate the volume of one 13\frac{1}{3}-meter cube

The volume of one cube with a side length of 13\frac{1}{3} meter is: Volume of one cube=(13)3=127cubic meters.\text{Volume of one cube} = \left( \frac{1}{3} \right)^3 = \frac{1}{27} \, \text{cubic meters}.

Step 3: Determine the number of 13\frac{1}{3}-meter cubes

To find the total number of cubes required, divide the volume of the rectangular prism by the volume of one cube: Number of cubes=Volume of prismVolume of one cube=89127.\text{Number of cubes} = \frac{\text{Volume of prism}}{\text{Volume of one cube}} = \frac{\frac{8}{9}}{\frac{1}{27}}. Perform the division by multiplying by the reciprocal of 127\frac{1}{27}: Number of cubes=89×27=8×279=2169=24.\text{Number of cubes} = \frac{8}{9} \times 27 = \frac{8 \times 27}{9} = \frac{216}{9} = 24.

Final Answer:

Robert will need 24 cubes to fill the box.


Would you like a detailed explanation of any step, or do you have further questions?

Here are 5 related questions to expand on this:

  1. How would the number of cubes change if the box dimensions were doubled?
  2. What is the total surface area of the rectangular prism?
  3. How do you calculate the surface area of one 13\frac{1}{3}-meter cube?
  4. How many 12\frac{1}{2}-meter cubes would fit into the same box?
  5. What is the significance of unit conversions when solving volume problems?

Tip: When working with fractions in geometry problems, always simplify intermediate steps to minimize errors.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Volume of a rectangular prism
Volume of a cube
Division of fractions

Formulas

Volume of rectangular prism = height × length × width
Volume of cube = side³
Number of cubes = Volume of prism ÷ Volume of one cube

Theorems

Principle of volume calculation for composite solids
Reciprocal rule for division of fractions

Suitable Grade Level

Grades 5-7