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Magazine Chapter 12: Problem 3
Describe real-world examples of five different solids. Write the formula used to find the volume of each type of solid.
Short Answer
Expert verified
Cube: dice; Rectangular Prism: shoebox; Sphere: basketball; Cylinder: soda can; Cone: ice cream cone.
Step by step solution
01
Identify a Cube
A common real-world example of a cube is a standard dice used in board games. Each side of a cube is square in shape.The formula to find the volume of a cube is given by: \[ V = a^3 \]where \( a \) is the length of a side of the cube.
02
Identify a Rectangular Prism
An example of a rectangular prism is a shoebox. All the angles are right angles, and has rectangular faces.The formula to find the volume of a rectangular prism is:\[ V = l \, w \, h \]where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
03
Identify a Sphere
A basketball or soccer ball is a real-world example of a sphere. A sphere has all points on its surface equidistant from its center.The volume formula for a sphere is:\[ V = \frac{4}{3} \pi r^3 \]where \( r \) is the radius of the sphere.
04
Identify a Cylinder
A soda can is a common example of a cylinder. It's a solid with two parallel circular bases and a curved surface connecting them.The formula used to find the volume of a cylinder is:\[ V = \pi r^2 h \]where \( r \) is the radius of the base and \( h \) is the height.
05
Identify a Cone
An ice cream cone is an example of a cone. A cone has a circular base that tapers smoothly up to a point called the apex.The formula to find the volume of a cone is:\[ V = \frac{1}{3} \pi r^2 h \]where \( r \) is the radius of the base and \( h \) is the height.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cube
A cube is a simple and fascinating solid shape, often seen in everyday items like dice or Rubik's cubes. What makes a cube stand out is its equal sides - each one is a square. Therefore, every face of the cube is the same size, and all angles are right angles.
A great way to picture a cube is to imagine a box where every edge is exactly the same length. To find the volume of a cube, you multiply the length of one side by itself twice. The formula is \[ V = a^3 \]Here, "a" represents the length of any side, as all sides are equal in a cube.
So next time you roll a die in a board game, you'll know the math behind its shape!
A great way to picture a cube is to imagine a box where every edge is exactly the same length. To find the volume of a cube, you multiply the length of one side by itself twice. The formula is \[ V = a^3 \]Here, "a" represents the length of any side, as all sides are equal in a cube.
So next time you roll a die in a board game, you'll know the math behind its shape!
Rectangular Prism
The rectangular prism is a versatile shape, resembling familiar objects like shoeboxes or packages. It has 6 faces, all of which are rectangles, making it unique compared to the perfect squares of a cube.
To calculate the volume of a rectangular prism, you need to know three things: length, width, and height. These dimensions can differ from one another, unlike a cube. The formula to determine the volume is \[ V = lwh \]
This formula simply multiplies all dimensions together to find how much space is inside. Practical, isn't it? Next time, while handling a shoebox, you'll understand its volume!
To calculate the volume of a rectangular prism, you need to know three things: length, width, and height. These dimensions can differ from one another, unlike a cube. The formula to determine the volume is \[ V = lwh \]
- "l" stands for length,
- "w" is the width, and
- "h" is the height.
This formula simply multiplies all dimensions together to find how much space is inside. Practical, isn't it? Next time, while handling a shoebox, you'll understand its volume!
Sphere
Spheres are smooth, round shapes that can be found in everyday items like basketballs or even the earth itself! What makes a sphere special is that each point on its surface is the same distance from the center.
The formula to find the volume of a sphere requires the radius, which is the distance from the center to the surface. The formula is \[ V = \frac{4}{3} \pi r^3 \]
Isn't it intriguing how this formula uses a fraction and a well-known mathematical constant to describe such a smooth shape?
The formula to find the volume of a sphere requires the radius, which is the distance from the center to the surface. The formula is \[ V = \frac{4}{3} \pi r^3 \]
- Here, "r" is the radius of the sphere.
- \( \pi \) is a constant that helps relate the diameter of a circle to its circumference.
Isn't it intriguing how this formula uses a fraction and a well-known mathematical constant to describe such a smooth shape?
Cylinder
Think about a soda can or a candle 鈥� these are common examples of cylinders. A cylinder is characterized by having two parallel circular bases and a curved surface that connects them.
To calculate the volume of a cylinder, you need the radius of the base and the height of the cylinder. The volume formula is \[ V = \pi r^2 h \]
This formula multiplies the area of the base by the cylinder's height, providing the space inside. It shows how a circle's two-dimensional area becomes a three-dimensional volume!
To calculate the volume of a cylinder, you need the radius of the base and the height of the cylinder. The volume formula is \[ V = \pi r^2 h \]
- "r" is the radius of the circular base,
- "h" is the height.
This formula multiplies the area of the base by the cylinder's height, providing the space inside. It shows how a circle's two-dimensional area becomes a three-dimensional volume!
Cone
Cones are common in everyday life; think of an ice cream cone or a party hat. A cone has a circular base that narrows smoothly up to a point called the apex.
Calculating the volume of a cone requires knowing the radius of the base and the height from the base to the apex. The formula is\[ V = \frac{1}{3} \pi r^2 h \]
This formula shows a cone's volume is exactly one-third of what a cylinder with the same base and height would be. One simple fraction changes how space is divided inside!
Calculating the volume of a cone requires knowing the radius of the base and the height from the base to the apex. The formula is\[ V = \frac{1}{3} \pi r^2 h \]
- "r" stands for the radius of the base,
- "h" is the height.
This formula shows a cone's volume is exactly one-third of what a cylinder with the same base and height would be. One simple fraction changes how space is divided inside!
