Isolating the Unknown Value z

Oct 13, 2025 by windailys.com 114 views

Introduction

In this article, we will explore the measures of the sides of a rectangular garden and use the given information to find the unknown value z. The measures of the sides are $\sqrt{3}z - 2$ feet and 7 feet, and the perimeter of the garden is 24 feet. We will use the formula for the perimeter of a rectangle, which is Perimeter = 2L + 2W, to write and solve an equation to find the unknown value z.

Understanding the Problem

The problem provides us with the measures of the sides of a rectangular garden, which are $\sqrt{3}z - 2$ feet and 7 feet. The perimeter of the garden is given as 24 feet. We need to use this information to find the unknown value z.

Writing the Equation

To find the unknown value z, we will use the formula for the perimeter of a rectangle, which is Perimeter = 2L + 2W. In this case, the perimeter is 24 feet, and the measures of the sides are $\sqrt{3}z - 2$ feet and 7 feet. We can write the equation as:

2( $\sqrt{3}z - 2$ ) + 2(7) = 24

Simplifying the Equation

To simplify the equation, we will start by multiplying the terms inside the parentheses:

2 $\sqrt{3}z - 4 + 14 = 24$

Next, we will combine like terms:

2 $\sqrt{3}z + 10 = 24$

Isolating the Unknown Value z

To isolate the unknown value z, we will subtract 10 from both sides of the equation:

2 $\sqrt{3}z = 14$

Solving for z

To solve for z, we will divide both sides of the equation by 2 $\sqrt{3}$ :

z = 14 / (2 $\sqrt{3}$ )

z = 7 / $\sqrt{3}$

Rationalizing the Denominator

To rationalize the denominator, we will multiply the numerator and denominator by $\sqrt{3}$ :

z = (7 / $\sqrt{3}$ ) * ( $\sqrt{3}$ / $\sqrt{3}$ )

z = 7 $\sqrt{3}$ / 3

Conclusion

In this article, we used the formula for the perimeter of a rectangle to write and solve an equation to find the unknown value z. The measures of the sides of the rectangular garden were $\sqrt{3}z - 2$ feet and 7 feet, and the perimeter was 24 feet. We simplified the equation, isolated the unknown value z, and solved for z. The final answer is z = 7 $\sqrt{3}$ / 3.

The Other Side

To find the other side of the rectangular garden, we can use the measures of the sides that we were given. The other side is 7 feet.

Discussion

This problem is a great example of how to use the formula for the perimeter of a rectangle to solve for an unknown value. The measures of the sides of the rectangular garden were given, and we used the formula to write and solve an equation to find the unknown value z. The final answer was z = 7 $\sqrt{3}$ / 3.

Mathematical Concepts

This problem involves the following mathematical concepts:

The formula for the perimeter of a rectangle
Simplifying equations
Isolating unknown values
Solving for unknown values
Rationalizing denominators

Real-World Applications

This problem has real-world applications in various fields, such as:

Architecture: When designing a building or a structure, architects need to consider the perimeter of the building to ensure that it is proportional to the size of the building.
Engineering: Engineers need to consider the perimeter of a structure when designing a system or a mechanism.
Landscaping: When designing a garden or a landscape, landscapers need to consider the perimeter of the garden to ensure that it is proportional to the size of the garden.

Conclusion

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

A: The formula for the perimeter of a rectangle is Perimeter = 2L + 2W, where L is the length and W is the width.

Q: How do I use the formula for the perimeter of a rectangle to solve for an unknown value?

A: To use the formula for the perimeter of a rectangle to solve for an unknown value, you need to plug in the values for the length and width into the formula and then solve for the unknown value.

Q: What is the unknown value z in the problem?

A: The unknown value z is the value that we are trying to find in the problem. In this case, the unknown value z is the value that we are trying to find in the equation 2( $\sqrt{3}z - 2$ ) + 2(7) = 24.

Q: How do I simplify the equation 2( $\sqrt{3}z - 2$ ) + 2(7) = 24?

A: To simplify the equation 2( $\sqrt{3}z - 2$ ) + 2(7) = 24, you need to multiply the terms inside the parentheses and then combine like terms.

Q: How do I isolate the unknown value z in the equation 2( $\sqrt{3}z - 2$ ) + 2(7) = 24?

A: To isolate the unknown value z in the equation 2( $\sqrt{3}z - 2$ ) + 2(7) = 24, you need to subtract 10 from both sides of the equation.

Q: How do I solve for the unknown value z in the equation 2( $\sqrt{3}z - 2$ ) + 2(7) = 24?

A: To solve for the unknown value z in the equation 2( $\sqrt{3}z - 2$ ) + 2(7) = 24, you need to divide both sides of the equation by 2 $\sqrt{3}$ .

Q: What is the final answer for the unknown value z?

A: The final answer for the unknown value z is z = 7 $\sqrt{3}$ / 3.

Q: What is the other side of the rectangular garden?

A: The other side of the rectangular garden is 7 feet.

Q: What are some real-world applications of the formula for the perimeter of a rectangle?

A: Some real-world applications of the formula for the perimeter of a rectangle include:

Architecture: When designing a building or a structure, architects need to consider the perimeter of the building to ensure that it is proportional to the size of the building.
Engineering: Engineers need to consider the perimeter of a structure when designing a system or a mechanism.
Landscaping: When designing a garden or a landscape, landscapers need to consider the perimeter of the garden to ensure that it is proportional to the size of the garden.

Q: What are some mathematical concepts that are used in this problem?

A: Some mathematical concepts that are used in this problem include:

The formula for the perimeter of a rectangle
Simplifying equations
Isolating unknown values
Solving for unknown values
Rationalizing denominators

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

A: Some tips for solving problems like this one include:

Read the problem carefully and understand what is being asked.
Use the formula for the perimeter of a rectangle to write an equation.
Simplify the equation and isolate the unknown value.
Solve for the unknown value using algebraic techniques.
Check your answer to make sure it is reasonable.