Karim Builds A Wooden Table.The Rectangular Top Of His Table Has An Area Of $3^7$ Square Inches And A Length Of $3^4$ Inches. What Is The Width Of The Table Top?A. 9 Inches B. 27 Inches C. 2,187 Inches D. 59,049 Inches

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Introduction

In this article, we will delve into the world of mathematics and solve a problem that involves finding the width of a rectangular table top. The problem is presented in a way that requires us to use our knowledge of exponents and algebra to find the solution. We will break down the problem step by step and provide a clear explanation of the solution.

Understanding the Problem

The problem states that the rectangular top of Karim's table has an area of $3^7$ square inches and a length of $3^4$ inches. We are asked to find the width of the table top. To solve this problem, we need to use the formula for the area of a rectangle, which is given by:

Area=length×width\text{Area} = \text{length} \times \text{width}

We are given the area and the length, so we can use this formula to find the width.

Breaking Down the Problem

Let's start by analyzing the given information. We know that the area of the table top is $3^7$ square inches, and the length is $3^4$ inches. We can write this information in the form of an equation:

37=34×width3^7 = 3^4 \times \text{width}

Now, we can use the properties of exponents to simplify this equation. We know that when we multiply two numbers with the same base, we can add their exponents. Therefore, we can rewrite the equation as:

37=34+1×width3^7 = 3^{4+1} \times \text{width}

This simplifies to:

37=35×width3^7 = 3^5 \times \text{width}

Solving for the Width

Now that we have simplified the equation, we can solve for the width. We can do this by dividing both sides of the equation by $3^5$:

3735=width\frac{3^7}{3^5} = \text{width}

Using the property of exponents that states $\frac{am}{an} = a^{m-n}$, we can simplify the left-hand side of the equation:

37−5=width3^{7-5} = \text{width}

This simplifies to:

32=width3^2 = \text{width}

Finding the Value of the Width

Now that we have found the value of the width in terms of an exponent, we can find the actual value by evaluating the exponent. We know that $3^2 = 9$, so we can substitute this value into the equation:

width=9\text{width} = 9

Therefore, the width of the table top is 9 inches.

Conclusion

In this article, we solved a problem that involved finding the width of a rectangular table top. We used the formula for the area of a rectangle and the properties of exponents to simplify the equation and find the value of the width. The final answer is 9 inches.

Answer Key

The correct answer is A. 9 inches.

Additional Tips and Tricks

  • When solving problems that involve exponents, make sure to use the properties of exponents to simplify the equation.
  • When dividing two numbers with the same base, subtract the exponents.
  • When multiplying two numbers with the same base, add the exponents.

Real-World Applications

This problem has real-world applications in various fields such as architecture, engineering, and design. For example, when designing a building, architects need to consider the area and dimensions of the rooms and spaces. Similarly, engineers need to consider the dimensions and area of the components and systems they design.

Final Thoughts

Q&A: Solving the Mystery of Karim's Wooden Table

In our previous article, we solved the problem of finding the width of Karim's wooden table. We used the formula for the area of a rectangle and the properties of exponents to simplify the equation and find the value of the width. In this article, we will provide a Q&A section to help readers understand the problem and its solution.

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is given by:

Area=length×width\text{Area} = \text{length} \times \text{width}

Q: How do we use the properties of exponents to simplify the equation?

A: When we multiply two numbers with the same base, we can add their exponents. Therefore, we can rewrite the equation as:

37=34+1×width3^7 = 3^{4+1} \times \text{width}

This simplifies to:

37=35×width3^7 = 3^5 \times \text{width}

Q: How do we solve for the width?

A: We can solve for the width by dividing both sides of the equation by $3^5$:

3735=width\frac{3^7}{3^5} = \text{width}

Using the property of exponents that states $\frac{am}{an} = a^{m-n}$, we can simplify the left-hand side of the equation:

37−5=width3^{7-5} = \text{width}

This simplifies to:

32=width3^2 = \text{width}

Q: What is the value of the width?

A: We know that $3^2 = 9$, so we can substitute this value into the equation:

width=9\text{width} = 9

Therefore, the width of the table top is 9 inches.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields such as architecture, engineering, and design. For example, when designing a building, architects need to consider the area and dimensions of the rooms and spaces. Similarly, engineers need to consider the dimensions and area of the components and systems they design.

Q: What are some tips and tricks for solving problems like this?

A: Here are some tips and tricks for solving problems like this:

  • When solving problems that involve exponents, make sure to use the properties of exponents to simplify the equation.
  • When dividing two numbers with the same base, subtract the exponents.
  • When multiplying two numbers with the same base, add the exponents.

Q: Can you provide some additional examples of problems like this?

A: Here are some additional examples of problems like this:

  • Find the area of a rectangle with a length of 6 inches and a width of 4 inches.
  • Find the width of a rectangle with an area of 12 square inches and a length of 3 inches.
  • Find the length of a rectangle with an area of 16 square inches and a width of 2 inches.

Conclusion

In this article, we provided a Q&A section to help readers understand the problem and its solution. We also provided some additional examples of problems like this and some tips and tricks for solving them. We hope this article has been helpful in understanding the problem and its solution.

Answer Key

The correct answer is A. 9 inches.

Additional Resources

For more information on exponents and algebra, please refer to the following resources:

  • Khan Academy: Exponents and Algebra
  • Mathway: Exponents and Algebra
  • Wolfram Alpha: Exponents and Algebra

Final Thoughts

In conclusion, solving the problem of finding the width of Karim's wooden table requires a clear understanding of the formula for the area of a rectangle and the properties of exponents. By breaking down the problem step by step and using the properties of exponents, we can find the value of the width. This problem has real-world applications and requires a clear understanding of mathematical concepts.