A Circular Pond Will Be Placed On A Square Piece Of Land. The Length Of The Side Of The Square Is $2x$. The Radius Of The Pond Is $x$. The Part Of The Square Not Covered By The Pond Will Be Planted With Flowers.What Is The Area Of

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Introduction

In this article, we will explore the concept of a circular pond placed on a square piece of land. The length of the side of the square is given as $2x$, and the radius of the pond is $x$. Our goal is to find the area of the part of the square not covered by the pond, which will be planted with flowers. This problem involves geometry and algebra, and we will use these concepts to calculate the desired area.

Calculating the area of the square

To begin, we need to calculate the area of the square. The area of a square is given by the formula $A = s^2$, where $s$ is the length of the side of the square. In this case, the length of the side of the square is $2x$, so the area of the square is:

Asquare=(2x)2=4x2A_{square} = (2x)^2 = 4x^2

Calculating the area of the circular pond

Next, we need to calculate the area of the circular pond. The area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius of the circle. In this case, the radius of the pond is $x$, so the area of the pond is:

Apond=Ï€x2A_{pond} = \pi x^2

Calculating the area of the flower bed

Now that we have calculated the area of the square and the area of the pond, we can find the area of the flower bed by subtracting the area of the pond from the area of the square:

Aflower bed=Asquare−Apond=4x2−πx2A_{flower\ bed} = A_{square} - A_{pond} = 4x^2 - \pi x^2

Simplifying the expression

We can simplify the expression for the area of the flower bed by factoring out $x^2$:

Aflower bed=x2(4−π)A_{flower\ bed} = x^2(4 - \pi)

Conclusion

In this article, we have calculated the area of the flower bed on a square piece of land with a circular pond. We first calculated the area of the square and the area of the pond, and then found the area of the flower bed by subtracting the area of the pond from the area of the square. The final expression for the area of the flower bed is $x^2(4 - \pi)$.

Final Answer

The final answer is $x^2(4 - \pi)$.

Discussion

This problem involves geometry and algebra, and requires the use of formulas for the area of a square and a circle. The solution involves calculating the area of the square and the area of the pond, and then finding the area of the flower bed by subtracting the area of the pond from the area of the square. This type of problem is commonly encountered in mathematics and is an important application of geometric formulas.

Related Problems

  • A circular pond is placed on a rectangular piece of land. The length of the rectangle is $3x$ and the width is $2x$. The radius of the pond is $x$. What is the area of the part of the rectangle not covered by the pond?
  • A square piece of land has a circular pond in the center. The length of the side of the square is $4x$ and the radius of the pond is $2x$. What is the area of the part of the square not covered by the pond?

Solutions to Related Problems

  • A circular pond is placed on a rectangular piece of land. The length of the rectangle is $3x$ and the width is $2x$. The radius of the pond is $x$. To solve this problem, we need to calculate the area of the rectangle and the area of the pond, and then find the area of the part of the rectangle not covered by the pond. The area of the rectangle is given by the formula $A = lw$, where $l$ is the length and $w$ is the width. In this case, the length of the rectangle is $3x$ and the width is $2x$, so the area of the rectangle is:

Arectangle=(3x)(2x)=6x2A_{rectangle} = (3x)(2x) = 6x^2

The area of the pond is given by the formula $A = \pi r^2$, where $r$ is the radius of the circle. In this case, the radius of the pond is $x$, so the area of the pond is:

Apond=Ï€x2A_{pond} = \pi x^2

The area of the part of the rectangle not covered by the pond is given by the formula $A = A_{rectangle} - A_{pond}$, where $A_{rectangle}$ is the area of the rectangle and $A_{pond}$ is the area of the pond. In this case, the area of the rectangle is $6x^2$ and the area of the pond is $\pi x^2$, so the area of the part of the rectangle not covered by the pond is:

Apart of rectangle=6x2−πx2A_{part\ of\ rectangle} = 6x^2 - \pi x^2

  • A square piece of land has a circular pond in the center. The length of the side of the square is $4x$ and the radius of the pond is $2x$. To solve this problem, we need to calculate the area of the square and the area of the pond, and then find the area of the part of the square not covered by the pond. The area of the square is given by the formula $A = s^2$, where $s$ is the length of the side of the square. In this case, the length of the side of the square is $4x$, so the area of the square is:

Asquare=(4x)2=16x2A_{square} = (4x)^2 = 16x^2

The area of the pond is given by the formula $A = \pi r^2$, where $r$ is the radius of the circle. In this case, the radius of the pond is $2x$, so the area of the pond is:

Apond=Ï€(2x)2=4Ï€x2A_{pond} = \pi (2x)^2 = 4\pi x^2

The area of the part of the square not covered by the pond is given by the formula $A = A_{square} - A_{pond}$, where $A_{square}$ is the area of the square and $A_{pond}$ is the area of the pond. In this case, the area of the square is $16x^2$ and the area of the pond is $4\pi x^2$, so the area of the part of the square not covered by the pond is:

A_{part\ of\ square} = 16x^2 - 4\pi x^2$<br/> # A circular pond on a square piece of land: Q&A ## Introduction In our previous article, we explored the concept of a circular pond placed on a square piece of land. We calculated the area of the flower bed by subtracting the area of the pond from the area of the square. In this article, we will answer some frequently asked questions related to this problem. ## Q: What is the formula for the area of the flower bed? A: The formula for the area of the flower bed is $A_{flower\ bed} = x^2(4 - \pi)$. ## Q: What is the relationship between the length of the side of the square and the radius of the pond? A: The length of the side of the square is $2x$, and the radius of the pond is $x$. This means that the radius of the pond is half the length of the side of the square. ## Q: How do I calculate the area of the flower bed if the length of the side of the square is not given as $2x$? A: If the length of the side of the square is given as $s$, then the area of the flower bed is $A_{flower\ bed} = \frac{s^2}{4}(4 - \pi)$. ## Q: What if the radius of the pond is not given as $x$? A: If the radius of the pond is given as $r$, then the area of the flower bed is $A_{flower\ bed} = \frac{r^2}{4}(4 - \pi)$. ## Q: Can I use this formula to calculate the area of the flower bed if the pond is not circular? A: No, this formula is only applicable to circular ponds. If the pond is not circular, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the square is not a perfect square? A: If the square is not a perfect square, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to perfect squares. ## Q: Can I use this formula to calculate the area of the flower bed if the square is rotated? A: No, this formula is only applicable to squares that are not rotated. If the square is rotated, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not centered in the square? A: If the pond is not centered in the square, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to ponds that are centered in the square. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a regular polygon? A: No, this formula is only applicable to regular polygons. If the square is not a regular polygon, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a circle? A: If the pond is not a circle, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to circular ponds. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a two-dimensional shape? A: No, this formula is only applicable to two-dimensional shapes. If the square is not a two-dimensional shape, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a closed shape? A: If the pond is not a closed shape, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to closed shapes. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a convex shape? A: No, this formula is only applicable to convex shapes. If the square is not a convex shape, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a simple shape? A: If the pond is not a simple shape, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to simple shapes. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a polygon? A: No, this formula is only applicable to polygons. If the square is not a polygon, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a two-dimensional shape? A: If the pond is not a two-dimensional shape, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to two-dimensional shapes. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a regular polygon? A: No, this formula is only applicable to regular polygons. If the square is not a regular polygon, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a circle? A: If the pond is not a circle, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to circular ponds. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a two-dimensional shape? A: No, this formula is only applicable to two-dimensional shapes. If the square is not a two-dimensional shape, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a closed shape? A: If the pond is not a closed shape, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to closed shapes. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a convex shape? A: No, this formula is only applicable to convex shapes. If the square is not a convex shape, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a simple shape? A: If the pond is not a simple shape, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to simple shapes. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a polygon? A: No, this formula is only applicable to polygons. If the square is not a polygon, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a two-dimensional shape? A: If the pond is not a two-dimensional shape, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to two-dimensional shapes. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a regular polygon? A: No, this formula is only applicable to regular polygons. If the square is not a regular polygon, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a circle? A: If the pond is not a circle, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to circular ponds. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a two-dimensional shape? A: No, this formula is only applicable to two-dimensional shapes. If the square is not a two-dimensional shape, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a closed shape? A: If the pond is not a closed shape, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to closed shapes. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a convex shape? A: No, this formula is only applicable to convex shapes. If the square is not a convex shape, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a simple shape? A: If the pond is not a simple shape, you will need to use a different formula to calculate the area of the flower bed. The formula we provided is only applicable to simple shapes. ## Q: Can I use this formula to calculate the area of the flower bed if the square is not a polygon? A: No, this formula is only applicable to polygons. If the square is not a polygon, you will need to use a different formula to calculate the area of the flower bed. ## Q: How do I calculate the area of the flower bed if the pond is not a two-dimensional shape