A Rectangle Has An Area Of $k^2+19k+60$ Square Inches. If The Value Of $k$ And The Dimensions Of The Rectangle Are All Natural Numbers, Which Statement About The Rectangle Could Be True?A. The Length Of The Rectangle Is

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Introduction

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In mathematics, a rectangle is a four-sided shape with opposite sides of equal length. The area of a rectangle is calculated by multiplying its length and width. In this article, we will explore the possibilities of a rectangle with a specified area, given by the expression $k^2+19k+60$ square inches. We will examine the conditions under which the value of $k$ and the dimensions of the rectangle are all natural numbers.

Understanding the Area Expression

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The given area expression is $k^2+19k+60$. To understand the possible values of $k$, we need to factorize the expression. Factoring the expression, we get:

$$k^2+19k+60 = (k+15)(k+4)$$

This means that the area of the rectangle can be expressed as the product of two factors, $(k+15)$ and $(k+4)$.

Conditions for Natural Number Dimensions

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For the dimensions of the rectangle to be natural numbers, both $(k+15)$ and $(k+4)$ must be natural numbers. This implies that $k+15 > 0$ and $k+4 > 0$. Solving these inequalities, we get:

$$k > -15$$

$$k > -4$$

Since $k$ is a natural number, we can conclude that $k \geq 1$.

Exploring Possible Values of $k$

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Now, let's explore the possible values of $k$ that satisfy the conditions. We know that $k \geq 1$ and $(k+15)(k+4)$ is a natural number. We can start by trying different values of $k$ and checking if the resulting area is a natural number.

Case 1: $k = 1$

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If $k = 1$, then the area expression becomes:

$$(1+15)(1+4) = 16 \times 5 = 80$$

This is a natural number, so $k = 1$ is a possible value.

Case 2: $k = 2$

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If $k = 2$, then the area expression becomes:

$$(2+15)(2+4) = 17 \times 6 = 102$$

This is a natural number, so $k = 2$ is a possible value.

Case 3: $k = 3$

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If $k = 3$, then the area expression becomes:

$$(3+15)(3+4) = 18 \times 7 = 126$$

This is a natural number, so $k = 3$ is a possible value.

Case 4: $k = 4$

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If $k = 4$, then the area expression becomes:

$$(4+15)(4+4) = 19 \times 8 = 152$$

This is a natural number, so $k = 4$ is a possible value.

Case 5: $k = 5$

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If $k = 5$, then the area expression becomes:

$$(5+15)(5+4) = 20 \times 9 = 180$$

This is a natural number, so $k = 5$ is a possible value.

Case 6: $k = 6$

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If $k = 6$, then the area expression becomes:

$$(6+15)(6+4) = 21 \times 10 = 210$$

This is a natural number, so $k = 6$ is a possible value.

Case 7: $k = 7$

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If $k = 7$, then the area expression becomes:

$$(7+15)(7+4) = 22 \times 11 = 242$$

This is a natural number, so $k = 7$ is a possible value.

Case 8: $k = 8$

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If $k = 8$, then the area expression becomes:

$$(8+15)(8+4) = 23 \times 12 = 276$$

This is a natural number, so $k = 8$ is a possible value.

Case 9: $k = 9$

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If $k = 9$, then the area expression becomes:

$$(9+15)(9+4) = 24 \times 13 = 312$$

This is a natural number, so $k = 9$ is a possible value.

Case 10: $k = 10$

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If $k = 10$, then the area expression becomes:

$$(10+15)(10+4) = 25 \times 14 = 350$$

This is a natural number, so $k = 10$ is a possible value.

Conclusion

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In conclusion, the possible values of $k$ that satisfy the conditions are $k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. For each of these values, the area of the rectangle is a natural number. Therefore, the statement about the rectangle could be true is:

The length of the rectangle is $k+15$ and the width is $k+4$, where $k$ is a natural number.

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Introduction

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In our previous article, we explored the possibilities of a rectangle with a specified area, given by the expression $k^2+19k+60$ square inches. We examined the conditions under which the value of $k$ and the dimensions of the rectangle are all natural numbers. In this article, we will answer some frequently asked questions about the rectangle.

Q&A

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Q: What is the area of the rectangle when $k = 1$?

A: When $k = 1$, the area of the rectangle is $(1+15)(1+4) = 16 \times 5 = 80$ square inches.

Q: What is the length and width of the rectangle when $k = 2$?

A: When $k = 2$, the length of the rectangle is $k+15 = 2+15 = 17$ inches and the width is $k+4 = 2+4 = 6$ inches.

Q: Is it possible to have a rectangle with an area of 100 square inches?

A: No, it is not possible to have a rectangle with an area of 100 square inches. The area expression is $k^2+19k+60$, and the only possible values of $k$ are $k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. For each of these values, the area is a natural number, but none of them are equal to 100.

Q: Can the dimensions of the rectangle be negative?

A: No, the dimensions of the rectangle cannot be negative. Since $k$ is a natural number, both $(k+15)$ and $(k+4)$ must be natural numbers. Therefore, the length and width of the rectangle must be positive.

Q: Is it possible to have a rectangle with an area of 200 square inches?

A: No, it is not possible to have a rectangle with an area of 200 square inches. The area expression is $k^2+19k+60$, and the only possible values of $k$ are $k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. For each of these values, the area is a natural number, but none of them are equal to 200.

Q: Can the value of $k$ be a fraction?

A: No, the value of $k$ cannot be a fraction. Since $k$ is a natural number, it must be an integer.

Q: Is it possible to have a rectangle with an area of 50 square inches?

A: No, it is not possible to have a rectangle with an area of 50 square inches. The area expression is $k^2+19k+60$, and the only possible values of $k$ are $k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. For each of these values, the area is a natural number, but none of them are equal to 50.

Conclusion

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In conclusion, we have answered some frequently asked questions about the rectangle with a specified area. We have shown that the possible values of $k$ are $k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$, and for each of these values, the area of the rectangle is a natural number. We have also shown that the dimensions of the rectangle cannot be negative, and the value of $k$ cannot be a fraction.

Additional Resources

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• Previous article: A Rectangle with a Specified Area
• Mathematics resources

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