The Area Of A Rectangular Field Is $14x^2y^3 + 21xy^2$ Square Meters.(a) If The Length Of The Field Is $7xy^2$ Meters, Find Its Breadth.(b) If $x = 3$ And $y = 1$, What Is The Actual Breadth And Area Of The Field?

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Introduction

In mathematics, the area of a rectangular field is a fundamental concept that is used to calculate the size of the field. The area of a rectangle is given by the product of its length and breadth. In this article, we will discuss how to find the breadth of a rectangular field given its area and length, and also calculate the actual area and breadth of the field when the values of the variables are given.

The Formula for the Area of a Rectangular Field

The area of a rectangular field is given by the formula:

Area = Length × Breadth

In this case, the area of the rectangular field is given as 14x2y3+21xy214x^2y^3 + 21xy^2 square meters, and the length of the field is given as 7xy27xy^2 meters. We need to find the breadth of the field.

Finding the Breadth of the Field

To find the breadth of the field, we can use the formula for the area of a rectangle and rearrange it to solve for the breadth. We have:

Area = Length × Breadth

Substituting the given values, we get:

14x2y3+21xy214x^2y^3 + 21xy^2 = 7xy27xy^2 × Breadth

To solve for the breadth, we can divide both sides of the equation by 7xy27xy^2:

Breadth = (14x2y3+21xy214x^2y^3 + 21xy^2) ÷ 7xy27xy^2

Simplifying the expression, we get:

Breadth = 2xy+3y2xy + 3y

Therefore, the breadth of the field is 2xy+3y2xy + 3y meters.

Calculating the Actual Area and Breadth of the Field

Now that we have found the expression for the breadth of the field, we can calculate the actual area and breadth of the field when the values of the variables are given. We are given that x=3x = 3 and y=1y = 1. Substituting these values into the expression for the breadth, we get:

Breadth = 2(3)(1)+3(1)2(3)(1) + 3(1) = 6+36 + 3 = 99 meters

Now that we have found the breadth of the field, we can calculate the actual area of the field by multiplying the length and breadth:

Area = Length × Breadth = 7(3)(1)27(3)(1)^2 × 99 = 189189 square meters

Therefore, the actual area of the field is 189189 square meters, and the actual breadth of the field is 99 meters.

Conclusion

In this article, we have discussed how to find the breadth of a rectangular field given its area and length, and also calculated the actual area and breadth of the field when the values of the variables are given. We have used the formula for the area of a rectangle and rearranged it to solve for the breadth. We have also substituted the given values into the expression for the breadth and calculated the actual area and breadth of the field. The actual area of the field is 189189 square meters, and the actual breadth of the field is 99 meters.

References

  • [1] "Mathematics for Engineers and Scientists" by Donald R. Hill
  • [2] "Calculus" by Michael Spivak

Further Reading

  • [1] "Geometry" by I.M. Gelfand
  • [2] "Algebra" by Michael Artin

Glossary

  • Area: The size of a two-dimensional region.
  • Breadth: The width of a rectangle.
  • Length: The distance between two opposite sides of a rectangle.
  • Rectangle: A four-sided shape with four right angles.
  • Variable: A symbol that represents a value that can change.
    The Area of a Rectangular Field: Q&A =====================================

Introduction

In our previous article, we discussed how to find the breadth of a rectangular field given its area and length, and also calculated the actual area and breadth of the field when the values of the variables are given. In this article, we will answer some frequently asked questions related to the area of a rectangular field.

Q: What is the formula for the area of a rectangular field?

A: The formula for the area of a rectangular field is:

Area = Length × Breadth

Q: How do I find the breadth of a rectangular field given its area and length?

A: To find the breadth of a rectangular field given its area and length, you can use the formula for the area of a rectangle and rearrange it to solve for the breadth. We have:

Area = Length × Breadth

Substituting the given values, we get:

Breadth = Area ÷ Length

Q: What is the difference between the area and the breadth of a rectangular field?

A: The area of a rectangular field is the size of the field, while the breadth is the width of the field. The area is calculated by multiplying the length and breadth, while the breadth is calculated by dividing the area by the length.

Q: Can I use the formula for the area of a rectangle to find the length of a rectangular field?

A: Yes, you can use the formula for the area of a rectangle to find the length of a rectangular field. We have:

Area = Length × Breadth

Substituting the given values, we get:

Length = Area ÷ Breadth

Q: What is the relationship between the area and the perimeter of a rectangular field?

A: The area of a rectangular field is the size of the field, while the perimeter is the distance around the field. The area is calculated by multiplying the length and breadth, while the perimeter is calculated by adding the lengths of all four sides.

Q: Can I use the formula for the area of a rectangle to find the perimeter of a rectangular field?

A: No, you cannot use the formula for the area of a rectangle to find the perimeter of a rectangular field. The perimeter is calculated by adding the lengths of all four sides, while the area is calculated by multiplying the length and breadth.

Q: What is the significance of the variables x and y in the formula for the area of a rectangular field?

A: The variables x and y in the formula for the area of a rectangular field represent the length and breadth of the field, respectively. The values of x and y can be used to calculate the actual area and breadth of the field.

Q: Can I use the formula for the area of a rectangle to find the area of a square?

A: Yes, you can use the formula for the area of a rectangle to find the area of a square. Since a square has equal length and breadth, the formula for the area of a square is:

Area = Length × Length = Length^2

Conclusion

In this article, we have answered some frequently asked questions related to the area of a rectangular field. We have discussed the formula for the area of a rectangle, how to find the breadth of a rectangular field given its area and length, and the relationship between the area and the perimeter of a rectangular field. We have also discussed the significance of the variables x and y in the formula for the area of a rectangular field.

References

  • [1] "Mathematics for Engineers and Scientists" by Donald R. Hill
  • [2] "Calculus" by Michael Spivak

Further Reading

  • [1] "Geometry" by I.M. Gelfand
  • [2] "Algebra" by Michael Artin

Glossary

  • Area: The size of a two-dimensional region.
  • Breadth: The width of a rectangle.
  • Length: The distance between two opposite sides of a rectangle.
  • Perimeter: The distance around a rectangle.
  • Rectangle: A four-sided shape with four right angles.
  • Square: A four-sided shape with four right angles and equal length and breadth.
  • Variable: A symbol that represents a value that can change.