/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 How many cubes measuring \(1 \ma... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 39

How many cubes measuring \(1 \mathrm{cm}\) on each edge will fit into the container? (figure not copy) A box measuring \(3 \mathrm{cm}\) by \(4 \mathrm{cm}\) by \(5 \mathrm{cm}\) on the inside edges

Short Answer

Expert verified
60 cubes

Step by step solution

01

Determine the Volume of the Box

Calculate the volume of the container using the formula for the volume of a rectangular prism: \Volume = \text{length} \times \text{width} \times \text{height}.\ Here, the dimensions are 3 cm (length), 4 cm (width), and 5 cm (height). So,\[ \text{Volume} = 3 \text{ cm} \times 4 \text{ cm} \times 5 \text{ cm} = 60 \text{ cm}^3 \]
02

Determine the Volume of a Small Cube

Calculate the volume of a cube with an edge length of 1 cm using the formula for the volume of a cube: \Volume = \text{edge length}^3.\ Here, the edge length is 1 cm. So,\[ \text{Volume} = 1 \text{ cm} \times 1 \text{ cm} \times 1 \text{ cm} = 1 \text{ cm}^3 = 1 \text{ cm}^3 \]
03

Calculate the Number of Small Cubes

Divide the volume of the container by the volume of one small cube to find how many small cubes will fit inside.\[ \text{Number of Cubes} = \frac{\text{Volume of the Box}}{\text{Volume of One Small Cube}} = \frac{60 \text{ cm}^3}{1 \text{ cm}^3} = 60 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Calculation of a Rectangular Prism
Calculating the volume of a rectangular prism is a fundamental concept in geometry. A rectangular prism is a three-dimensional shape with six faces, all of which are rectangles. To find its volume, you multiply its length, width, and height. The formula is: \(\text{Volume} = \text{length} \times \text{width} \times \text{height}\). In our exercise, the dimensions provided are 3 cm (length), 4 cm (width), and 5 cm (height). Plugging these into the formula, we get: \[ \text{Volume} = 3 \text{ cm} \times 4 \text{ cm} \times 5 \text{ cm} = 60 \text{ cm}^3 \]. This volume calculation tells us the total space inside the container.
Calculating Cube Volume
Next, let's understand how to find the volume of a cube. A cube is a special type of rectangular prism where all edges are the same length. To find its volume, you use the formula: \(\text{Volume} = \text{edge length}^3\). In our problem, the edge length is 1 cm. So, the calculation becomes: \[ \text{Volume} = 1 \text{ cm} \times 1 \text{ cm} \times 1 \text{ cm} = 1 \text{ cm}^3 \]. This tells us that each small cube occupies 1 cubic centimeter of space.
Volume Division to Determine Number of Cubes
To discover how many small cubes will fit inside the larger container, we need to perform volume division. This means we divide the volume of the rectangular prism by the volume of one small cube. Using our earlier calculations: \[ \text{Number of Cubes} = \frac{\text{Volume of the Box}}{\text{Volume of One Small Cube}} = \frac{60 \text{ cm}^3}{1 \text{ cm}^3} = 60 \]. This division tells us that 60 small cubes, each measuring 1 cm on each edge, can fit into the container with dimensions 3 cm by 4 cm by 5 cm. Breaking down larger volumes into smaller, manageable parts is essential in many real-life applications such as packing, storage, and resource management.

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Most popular questions from this chapter

Lickety Split ice cream comes in a cylindrical container with an inside diameter of 6 inches and a height of 10 inches. The company claims to give the customer 25 scoops of ice cream per container, each scoop being a sphere with a 3 -inch diameter. How many scoops will each container really hold?

Identify each statement as true or false. Sketch a counterexample for each false statement or explain why it is false. Every slice of a prism cut parallel to the bases is congruent to the bases.

Application A farmer must periodically resurface the interior (wall, floor, and ceiling) of his silo to protect it from the acid created by the silage. The height of the silo to the top of the hemispherical dome is \(50 \mathrm{ft}\), and the diameter is \(18 \mathrm{ft}\). a. What is the approximate surface area that needs to be treated? b. If 1 gallon of resurfacing compound covers about \(250 \mathrm{ft}^{2}\) how many gallons are needed? c. There is 0.8 bushel per \(\mathrm{ft}^{3}\). Calculate the number of bushels of grain this silo will hold. (IMAGE CAN'T COPY)

A fish tank 10 by 14 by 12 inches high is the home of a large goldfish named Columbia. She is taken out when her owner cleans the tank, and the water level in the tank drops \(\frac{1}{3}\) inch. What is Columbia's volume?

As bad as tanker oil spills are, they are only about \(12 \%\) the 3.5 million tons of oil that enters the oceans each year. The rest comes from routine tanker operations, sewage treatment plants' runoff, natural sources, and offshore oil rigs. One month's maintenance and routine operation of a single sypertanker produces up to \(17,000\) gallons of oil sludge that gets into the ocean! If a cylindrical barrel is about 1.6 feet in diameter and 2.8 feet tall, how many barrels are needed to hold \(17,000\) gallons of oil sludge? Recall that a cubic foot water is about 7.5 gallons.
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