A Rectangle Is Inscribed With Its Base On The $x − A X I S -axis − A X I S And Its Upper Corners On The Parabola $y=7-x^2$. What Are The Dimensions Of Such A Rectangle With The Greatest Possible Area?Width = □ = \square = □ Height = □ = \square = □

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Introduction

In this article, we will explore the problem of finding the dimensions of a rectangle inscribed in a parabola with the greatest possible area. The parabola is given by the equation y=7−x2y=7-x^2, and the rectangle's base lies on the xx-axis. We will use calculus to find the maximum area of the rectangle and determine its dimensions.

Problem Statement

The problem can be stated as follows:

  • Find the dimensions of a rectangle with its base on the xx-axis and its upper corners on the parabola y=7−x2y=7-x^2 that has the greatest possible area.

Background

To solve this problem, we need to understand the concept of optimization and how to use calculus to find the maximum or minimum value of a function. In this case, we want to find the maximum area of the rectangle, which is a function of the width and height of the rectangle.

Solution Strategy

Our solution strategy will be to:

  1. Define the area of the rectangle as a function of the width and height.
  2. Use calculus to find the maximum value of the area function.
  3. Determine the dimensions of the rectangle that correspond to the maximum area.

Step 1: Define the Area Function

Let's define the area of the rectangle as a function of the width and height. Since the base of the rectangle lies on the xx-axis, the width of the rectangle is equal to the distance between the two points where the rectangle intersects the xx-axis. Let's call this distance 2x2x. The height of the rectangle is equal to the distance between the xx-axis and the point where the rectangle intersects the parabola. Let's call this distance yy.

The area of the rectangle is given by the formula:

A(x)=2x⋅yA(x) = 2x \cdot y

Substituting the equation of the parabola, we get:

A(x)=2x⋅(7−x2)A(x) = 2x \cdot (7-x^2)

Step 2: Find the Maximum Value of the Area Function

To find the maximum value of the area function, we need to take the derivative of the area function with respect to xx and set it equal to zero.

dAdx=2(7−x2)+2x⋅(−2x)\frac{dA}{dx} = 2(7-x^2) + 2x \cdot (-2x)

Simplifying the expression, we get:

dAdx=14−4x2−4x2\frac{dA}{dx} = 14 - 4x^2 - 4x^2

dAdx=14−8x2\frac{dA}{dx} = 14 - 8x^2

Setting the derivative equal to zero, we get:

14−8x2=014 - 8x^2 = 0

Solving for xx, we get:

x2=148x^2 = \frac{14}{8}

x2=74x^2 = \frac{7}{4}

x=±74x = \pm \sqrt{\frac{7}{4}}

Since the width of the rectangle is equal to 2x2x, we get:

Width=2⋅74\text{Width} = 2 \cdot \sqrt{\frac{7}{4}}

Width=7\text{Width} = \sqrt{7}

Step 3: Determine the Dimensions of the Rectangle

Now that we have found the width of the rectangle, we can determine the height of the rectangle by substituting the value of xx into the equation of the parabola.

y=7−x2y = 7 - x^2

y=7−(74)2y = 7 - \left(\sqrt{\frac{7}{4}}\right)^2

y=7−74y = 7 - \frac{7}{4}

y=214y = \frac{21}{4}

So, the dimensions of the rectangle with the greatest possible area are:

  • Width: 7\boxed{\sqrt{7}}
  • Height: 214\boxed{\frac{21}{4}}

Conclusion

In this article, we have used calculus to find the dimensions of a rectangle inscribed in a parabola with the greatest possible area. We have defined the area function, taken the derivative of the area function, and set it equal to zero to find the maximum value of the area function. We have then determined the dimensions of the rectangle that correspond to the maximum area. The dimensions of the rectangle are:

  • Width: 7\boxed{\sqrt{7}}
  • Height: 214\boxed{\frac{21}{4}}

These dimensions represent the maximum possible area of a rectangle inscribed in the given parabola.

References

  • [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
  • [2] Mathematics for Computer Science, Eric Lehman, F Thomson Leighton, and Albert R Meyer, 2001.

Discussion

This problem is a classic example of an optimization problem, where we want to find the maximum or minimum value of a function. In this case, we want to find the maximum area of a rectangle inscribed in a parabola. The solution involves using calculus to find the maximum value of the area function and determining the dimensions of the rectangle that correspond to the maximum area.

The problem can be solved using various methods, including the method of Lagrange multipliers or the method of substitution. However, the method of substitution is the most straightforward and easiest to understand.

The problem has many real-world applications, such as finding the maximum area of a rectangle that can be inscribed in a given shape or finding the maximum volume of a container that can be inscribed in a given shape.

Related Problems

  • Find the dimensions of a rectangle with its base on the xx-axis and its upper corners on the parabola y=4−x2y=4-x^2 that has the greatest possible area.
  • Find the dimensions of a rectangle with its base on the yy-axis and its upper corners on the parabola x=y2x=y^2 that has the greatest possible area.
  • Find the dimensions of a rectangle with its base on the xx-axis and its upper corners on the ellipse x24+y29=1\frac{x^2}{4} + \frac{y^2}{9} = 1 that has the greatest possible area.

These problems are similar to the problem we have solved in this article and can be solved using the same method.

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Introduction

In our previous article, we explored the problem of finding the dimensions of a rectangle inscribed in a parabola with the greatest possible area. We used calculus to find the maximum value of the area function and determined the dimensions of the rectangle that correspond to the maximum area. In this article, we will answer some of the most frequently asked questions related to this problem.

Q&A

Q1: What is the equation of the parabola?

A1: The equation of the parabola is y=7−x2y=7-x^2.

Q2: What is the width of the rectangle with the greatest possible area?

A2: The width of the rectangle with the greatest possible area is 7\boxed{\sqrt{7}}.

Q3: What is the height of the rectangle with the greatest possible area?

A3: The height of the rectangle with the greatest possible area is 214\boxed{\frac{21}{4}}.

Q4: How did you find the maximum value of the area function?

A4: We found the maximum value of the area function by taking the derivative of the area function with respect to xx and setting it equal to zero.

Q5: What is the significance of the point where the rectangle intersects the parabola?

A5: The point where the rectangle intersects the parabola is the point where the height of the rectangle is equal to the distance between the xx-axis and the point of intersection.

Q6: Can you explain the concept of optimization in this problem?

A6: Optimization in this problem refers to finding the maximum or minimum value of a function, in this case, the area function. We used calculus to find the maximum value of the area function.

Q7: What are some real-world applications of this problem?

A7: Some real-world applications of this problem include finding the maximum area of a rectangle that can be inscribed in a given shape or finding the maximum volume of a container that can be inscribed in a given shape.

Q8: Can you provide some examples of similar problems?

A8: Some examples of similar problems include:

  • Find the dimensions of a rectangle with its base on the xx-axis and its upper corners on the parabola y=4−x2y=4-x^2 that has the greatest possible area.
  • Find the dimensions of a rectangle with its base on the yy-axis and its upper corners on the parabola x=y2x=y^2 that has the greatest possible area.
  • Find the dimensions of a rectangle with its base on the xx-axis and its upper corners on the ellipse x24+y29=1\frac{x^2}{4} + \frac{y^2}{9} = 1 that has the greatest possible area.

Conclusion

In this article, we have answered some of the most frequently asked questions related to the problem of finding the dimensions of a rectangle inscribed in a parabola with the greatest possible area. We have provided explanations and examples to help clarify the concepts and methods used in the solution.

References

  • [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
  • [2] Mathematics for Computer Science, Eric Lehman, F Thomson Leighton, and Albert R Meyer, 2001.

Discussion

This problem is a classic example of an optimization problem, where we want to find the maximum or minimum value of a function. In this case, we want to find the maximum area of a rectangle inscribed in a parabola. The solution involves using calculus to find the maximum value of the area function and determining the dimensions of the rectangle that correspond to the maximum area.

The problem has many real-world applications, such as finding the maximum area of a rectangle that can be inscribed in a given shape or finding the maximum volume of a container that can be inscribed in a given shape.

Related Problems

  • Find the dimensions of a rectangle with its base on the xx-axis and its upper corners on the parabola y=4−x2y=4-x^2 that has the greatest possible area.
  • Find the dimensions of a rectangle with its base on the yy-axis and its upper corners on the parabola x=y2x=y^2 that has the greatest possible area.
  • Find the dimensions of a rectangle with its base on the xx-axis and its upper corners on the ellipse x24+y29=1\frac{x^2}{4} + \frac{y^2}{9} = 1 that has the greatest possible area.

These problems are similar to the problem we have solved in this article and can be solved using the same method.