Given That The Area Of A Rectangular Flower Garden Is $(2x+3)(y-5)$ Cm\[$^2\$\] And The Width Is $(2y-10)$ Cm, Find Its Length.Recall That The Area $A = L \times W$, Where \$L$[/tex\] Is The Length

Feb 26, 2025 by ADMIN 205 views

**Finding the Length of a Rectangular Flower Garden**

Introduction

When it comes to calculating the dimensions of a rectangular flower garden, we often need to use the formula for the area of a rectangle, which is given by $A = L \times W$ , where $A$ is the area, $L$ is the length, and $W$ is the width. In this article, we will use this formula to find the length of a rectangular flower garden given its area and width.

Given Information

The area of the rectangular flower garden is given by the expression $(2x+3)(y-5)$ cm $^2$ , and the width is given by the expression $(2y-10)$ cm. We are asked to find the length of the garden.

Recalling the Formula for the Area of a Rectangle

The formula for the area of a rectangle is given by $A = L \times W$ , where $A$ is the area, $L$ is the length, and $W$ is the width. We can use this formula to find the length of the garden by rearranging it to solve for $L$ .

Rearranging the Formula to Solve for L

We can rearrange the formula $A = L \times W$ to solve for $L$ by dividing both sides by $W$ . This gives us the formula $L = \frac{A}{W}$ .

Substituting the Given Values

We can substitute the given values into the formula $L = \frac{A}{W}$ . The area $A$ is given by the expression $(2x+3)(y-5)$ cm $^2$ , and the width $W$ is given by the expression $(2y-10)$ cm.

Simplifying the Expression

We can simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of $(2y-10)$ is $(2y+10)$ .

Multiplying the Numerator and Denominator by the Conjugate

We can multiply the numerator and denominator by the conjugate $(2y+10)$ to get:

L = \frac{(2x+3)(y-5)(2y+10)}{(2y-10)(2y+10)}

Simplifying the Expression Further

We can simplify the expression further by canceling out the common factors in the numerator and denominator.

Canceling Out Common Factors

We can cancel out the common factors $(2y-10)$ and $(2y+10)$ in the numerator and denominator to get:

L = \frac{(2x+3)(y-5)}{2}

Simplifying the Expression Even Further

We can simplify the expression even further by multiplying the numerator and denominator by $2$ to get:

L = \frac{(2x+3)(y-5) \times 2}{2 \times 2}

Simplifying the Expression to Get the Final Answer

We can simplify the expression to get the final answer:

L = \frac{(2x+3)(y-5)}{2}

Conclusion

In this article, we used the formula for the area of a rectangle to find the length of a rectangular flower garden given its area and width. We started by recalling the formula $A = L \times W$ , and then rearranged it to solve for $L$ . We substituted the given values into the formula and simplified the expression to get the final answer.

Final Answer

The final answer is $\boxed{\frac{(2x+3)(y-5)}{2}}$ .

Discussion

This problem is a great example of how to use the formula for the area of a rectangle to find the length of a rectangular flower garden. It requires the student to recall the formula, rearrange it to solve for $L$ , and then substitute the given values into the formula. The student must also simplify the expression to get the final answer.

Tips and Tricks

Make sure to recall the formula for the area of a rectangle before starting the problem.
Rearrange the formula to solve for $L$ by dividing both sides by $W$ .
Substitute the given values into the formula and simplify the expression to get the final answer.
Make sure to cancel out common factors in the numerator and denominator.

Common Mistakes

Forgetting to recall the formula for the area of a rectangle.
Not rearranging the formula to solve for $L$ .
Not simplifying the expression to get the final answer.
Not canceling out common factors in the numerator and denominator.

Real-World Applications

This problem has real-world applications in gardening and landscaping. When designing a rectangular flower garden, it is essential to know the length and width of the garden to ensure that it is proportional and aesthetically pleasing.

Extension

This problem can be extended by adding more variables to the formula. For example, we could add a third variable $z$ to the formula to represent the depth of the garden. This would require the student to use the formula $A = L \times W \times D$ to find the length of the garden.

Conclusion

In conclusion, this problem is a great example of how to use the formula for the area of a rectangle to find the length of a rectangular flower garden. It requires the student to recall the formula, rearrange it to solve for $L$ , and then substitute the given values into the formula. The student must also simplify the expression to get the final answer.

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

A: The formula for the area of a rectangle is given by $A = L \times W$ , where $A$ is the area, $L$ is the length, and $W$ is the width.

Q: How do I find the length of a rectangular flower garden given its area and width?

A: To find the length of a rectangular flower garden given its area and width, you can use the formula $L = \frac{A}{W}$ , where $A$ is the area and $W$ is the width.

Q: What if the area is given by an expression involving variables?

A: If the area is given by an expression involving variables, you can substitute the expression into the formula $L = \frac{A}{W}$ and simplify to find the length.

Q: How do I simplify the expression to find the length?

A: To simplify the expression, you can cancel out common factors in the numerator and denominator, and then multiply the numerator and denominator by the conjugate of the denominator.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a + b$ is given by $a - b$ .

Q: How do I use the conjugate to simplify the expression?

A: To use the conjugate to simplify the expression, you can multiply the numerator and denominator by the conjugate of the denominator.

Q: What if the width is given by an expression involving variables?

A: If the width is given by an expression involving variables, you can substitute the expression into the formula $L = \frac{A}{W}$ and simplify to find the length.

Q: How do I find the length of a rectangular flower garden with a variable width?

A: To find the length of a rectangular flower garden with a variable width, you can use the formula $L = \frac{A}{W}$ , where $A$ is the area and $W$ is the width.

Q: What if the area and width are both given by expressions involving variables?

A: If the area and width are both given by expressions involving variables, you can substitute the expressions into the formula $L = \frac{A}{W}$ and simplify to find the length.

Q: How do I simplify the expression when both the area and width are given by expressions involving variables?

A: To simplify the expression when both the area and width are given by expressions involving variables, you can cancel out common factors in the numerator and denominator, and then multiply the numerator and denominator by the conjugate of the denominator.

Q: What are some common mistakes to avoid when finding the length of a rectangular flower garden?

A: Some common mistakes to avoid when finding the length of a rectangular flower garden include forgetting to recall the formula for the area of a rectangle, not rearranging the formula to solve for $L$ , and not simplifying the expression to get the final answer.

Q: What are some real-world applications of finding the length of a rectangular flower garden?

A: Some real-world applications of finding the length of a rectangular flower garden include designing a rectangular flower garden, calculating the area of a rectangular flower garden, and determining the length of a rectangular flower garden.

Q: How can I extend this problem to include more variables?

A: You can extend this problem to include more variables by adding a third variable $z$ to the formula to represent the depth of the garden. This would require the student to use the formula $A = L \times W \times D$ to find the length of the garden.

Q: What are some tips and tricks for finding the length of a rectangular flower garden?

A: Some tips and tricks for finding the length of a rectangular flower garden include recalling the formula for the area of a rectangle, rearranging the formula to solve for $L$ , and simplifying the expression to get the final answer.

Q: What are some common misconceptions about finding the length of a rectangular flower garden?

A: Some common misconceptions about finding the length of a rectangular flower garden include thinking that the length is always equal to the width, and thinking that the area is always equal to the length times the width.

Q: How can I use technology to find the length of a rectangular flower garden?

A: You can use technology such as calculators or computer software to find the length of a rectangular flower garden by plugging in the values for the area and width and solving for the length.

Q: What are some real-world examples of finding the length of a rectangular flower garden?

A: Some real-world examples of finding the length of a rectangular flower garden include designing a rectangular flower garden for a client, calculating the area of a rectangular flower garden for a landscaping project, and determining the length of a rectangular flower garden for a gardening competition.