A Right Pyramid With A Square Base Has A Base Length Of $x$ Inches, And The Height Is Two Inches Longer Than The Length Of The Base. Which Expression Represents The Volume In Terms Of $x$?A. X 2 ( X + 2 ) 3 \frac{x^2(x+2)}{3} 3 X 2 ( X + 2 ) ​ Cubic

by ADMIN 254 views

In this problem, we are tasked with finding the expression that represents the volume of a right pyramid with a square base. The base length of the pyramid is given as $x$ inches, and the height is two inches longer than the length of the base. To find the volume, we need to use the formula for the volume of a pyramid, which is given by $V = \frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height.

Calculating the Height of the Pyramid

The height of the pyramid is two inches longer than the length of the base, which is $x$ inches. Therefore, the height can be represented as $x + 2$ inches.

Calculating the Area of the Base

The base of the pyramid is a square with a length of $x$ inches. The area of a square is given by the formula $A = s^2$, where $s$ is the length of the side. In this case, the area of the base is $x^2$ square inches.

Calculating the Volume of the Pyramid

Now that we have the area of the base and the height, we can use the formula for the volume of a pyramid to find the expression that represents the volume in terms of $x$. Plugging in the values, we get:

V=13BhV = \frac{1}{3}Bh

V=13x2(x+2)V = \frac{1}{3}x^2(x + 2)

Simplifying the expression, we get:

V=x2(x+2)3V = \frac{x^2(x + 2)}{3}

Conclusion

In this problem, we were tasked with finding the expression that represents the volume of a right pyramid with a square base. We used the formula for the volume of a pyramid and the given information about the base length and height to find the expression. The final expression for the volume in terms of $x$ is $\frac{x^2(x + 2)}{3}$ cubic inches.

Discussion

This problem requires a good understanding of the formula for the volume of a pyramid and the ability to apply it to a given situation. It also requires the ability to simplify expressions and solve for the unknown variable. The problem is a good example of how to use mathematical formulas to solve real-world problems.

Related Topics

  • Volume of a pyramid
  • Area of a square
  • Simplifying expressions
  • Solving for unknown variables

Example Problems

  • Find the volume of a pyramid with a square base and a height of 5 inches.
  • Find the area of a square with a side length of 4 inches.
  • Simplify the expression $\frac{x^2 + 2x}{x + 1}$.
  • Solve for the unknown variable in the equation $2x + 5 = 11$.

Practice Problems

  • Find the volume of a pyramid with a square base and a height of 3 inches.
  • Find the area of a square with a side length of 6 inches.
  • Simplify the expression $\frac{x^2 - 2x}{x - 1}$.
  • Solve for the unknown variable in the equation $3x - 2 = 7$.

Glossary

  • Pyramid: A three-dimensional shape with a square or triangular base and four triangular faces that meet at the apex.
  • Volume: The amount of space inside a three-dimensional shape.
  • Area: The amount of space inside a two-dimensional shape.
  • Simplifying expressions: Reducing a complex expression to its simplest form.
  • Solving for unknown variables: Finding the value of an unknown variable in an equation.
    Q&A: Understanding the Volume of a Right Pyramid with a Square Base ====================================================================

In the previous article, we discussed how to find the expression that represents the volume of a right pyramid with a square base. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the formula for the volume of a pyramid?

A: The formula for the volume of a pyramid is given by $V = \frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height.

Q: How do I find the area of the base of a pyramid?

A: To find the area of the base of a pyramid, you need to know the length of the side of the base. The area of a square is given by the formula $A = s^2$, where $s$ is the length of the side.

Q: What is the relationship between the base length and the height of a pyramid?

A: In the problem we discussed earlier, the height of the pyramid is two inches longer than the length of the base. This means that if the base length is $x$ inches, the height is $x + 2$ inches.

Q: How do I simplify the expression for the volume of a pyramid?

A: To simplify the expression for the volume of a pyramid, you need to multiply the terms together and then combine like terms. For example, the expression $\frac{x^2(x + 2)}{3}$ can be simplified by multiplying the terms together and then combining like terms.

Q: What is the final expression for the volume of a right pyramid with a square base?

A: The final expression for the volume of a right pyramid with a square base is $\frac{x^2(x + 2)}{3}$ cubic inches.

Q: How do I use the formula for the volume of a pyramid to solve real-world problems?

A: To use the formula for the volume of a pyramid to solve real-world problems, you need to know the length of the side of the base and the height of the pyramid. You can then plug these values into the formula to find the volume of the pyramid.

Q: What are some common mistakes to avoid when finding the volume of a pyramid?

A: Some common mistakes to avoid when finding the volume of a pyramid include:

  • Forgetting to multiply the terms together when simplifying the expression
  • Forgetting to combine like terms when simplifying the expression
  • Using the wrong formula for the volume of a pyramid
  • Not knowing the length of the side of the base or the height of the pyramid

Q: How do I check my work when finding the volume of a pyramid?

A: To check your work when finding the volume of a pyramid, you can plug the values into the formula and then simplify the expression. You can also use a calculator to check your work.

Q: What are some real-world applications of the formula for the volume of a pyramid?

A: Some real-world applications of the formula for the volume of a pyramid include:

  • Finding the volume of a pyramid-shaped container
  • Finding the volume of a pyramid-shaped building
  • Finding the volume of a pyramid-shaped object

Q: How do I find the volume of a pyramid with a triangular base?

A: To find the volume of a pyramid with a triangular base, you need to know the length of the base and the height of the pyramid. You can then use the formula for the volume of a pyramid to find the volume.

Q: What is the difference between the volume of a pyramid and the volume of a cone?

A: The volume of a pyramid is given by the formula $V = \frac{1}{3}Bh$, while the volume of a cone is given by the formula $V = \frac{1}{3}\pi r^2h$, where $r$ is the radius of the base. The main difference between the two formulas is the presence of the $\pi$ term in the formula for the volume of a cone.

Q: How do I find the volume of a pyramid with a rectangular base?

A: To find the volume of a pyramid with a rectangular base, you need to know the length and width of the base and the height of the pyramid. You can then use the formula for the volume of a pyramid to find the volume.

Q: What are some common real-world objects that are shaped like pyramids?

A: Some common real-world objects that are shaped like pyramids include:

  • The Great Pyramid of Giza
  • The Washington Monument
  • A pyramid-shaped container
  • A pyramid-shaped building

Q: How do I use the formula for the volume of a pyramid to solve problems involving similar figures?

A: To use the formula for the volume of a pyramid to solve problems involving similar figures, you need to know the length of the side of the base and the height of the pyramid. You can then use the formula to find the volume of the pyramid and then use the concept of similar figures to find the volume of the other pyramid.