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Maximum area of a rectangle which can be inscribed in a circle of given radius R is given by αR2. Find the value of α.
Correct answer is '2'. Can you explain this answer?
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Maximum area of a rectangle which can be inscribed in a circle of give...
Let rectangle has width b and height h.
Area = h·b
Also, b2 + h2 = (2R)2 = 4R2
Area is maximum when A2 is maxima
A2 = h2(4R2 – h2)]
f(h) = h2(4R2 – h2)
For maxima,
⇒
= h2(–2h) + (4R2 – h2)2h = 0
Area = h·b
Also, b2 + h2 = (2R)2 = 4R2
Area is maximum when A2 is maxima
A2 = h2(4R2 – h2)]
f(h) = h2(4R2 – h2)
For maxima,
⇒
= h2(–2h) + (4R2 – h2)2h = 0–h2 + 4R2 – h2 = 0
h2 = 2R2
h = √2
h2 = 2R2
h = √2
From physical nature of problem, it is clear that this should be maximum area since minimum area will tend towards zero.
Hence, value of α = 2
The correct answer is: 2
Hence, value of α = 2
The correct answer is: 2
Most Upvoted Answer
Maximum area of a rectangle which can be inscribed in a circle of give...
Understanding the Problem
To find the maximum area of a rectangle inscribed in a circle with a given radius R, we can use geometric principles.
Circle and Rectangle Relationship
- The rectangle will be inscribed in the circle such that all four corners touch the circumference.
- The diameter of the circle serves as the diagonal of the rectangle.
Dimensions of the Rectangle
- Let the rectangle have length l and width w.
- The relationship between the rectangle's dimensions and the circle's radius can be expressed using the Pythagorean theorem:
l² + w² = (2R)².
Area of the Rectangle
- The area A of the rectangle is given by:
A = l * w.
Maximizing the Area
- To maximize A, we express w in terms of l:
w = sqrt((2R)² - l²).
- Substitute w back into the area formula:
A(l) = l * sqrt((2R)² - l²).
Finding the Maximum Area
- To find the maximum, take the derivative of A with respect to l, set it to zero, and solve.
- The maximum area occurs when l = w, meaning the rectangle is a square.
Calculating Maximum Area
- When l = w, we have:
A = l * l = l².
- Using the relationship l² + l² = (2R)², we find:
2l² = 4R², leading to l² = 2R².
Final Area Value
- Thus, the maximum area A = l² = 2R².
- This shows that the maximum area of the inscribed rectangle is given by αR², where α = 2.
In conclusion, the value of α is 2.
To find the maximum area of a rectangle inscribed in a circle with a given radius R, we can use geometric principles.
Circle and Rectangle Relationship
- The rectangle will be inscribed in the circle such that all four corners touch the circumference.
- The diameter of the circle serves as the diagonal of the rectangle.
Dimensions of the Rectangle
- Let the rectangle have length l and width w.
- The relationship between the rectangle's dimensions and the circle's radius can be expressed using the Pythagorean theorem:
l² + w² = (2R)².
Area of the Rectangle
- The area A of the rectangle is given by:
A = l * w.
Maximizing the Area
- To maximize A, we express w in terms of l:
w = sqrt((2R)² - l²).
- Substitute w back into the area formula:
A(l) = l * sqrt((2R)² - l²).
Finding the Maximum Area
- To find the maximum, take the derivative of A with respect to l, set it to zero, and solve.
- The maximum area occurs when l = w, meaning the rectangle is a square.
Calculating Maximum Area
- When l = w, we have:
A = l * l = l².
- Using the relationship l² + l² = (2R)², we find:
2l² = 4R², leading to l² = 2R².
Final Area Value
- Thus, the maximum area A = l² = 2R².
- This shows that the maximum area of the inscribed rectangle is given by αR², where α = 2.
In conclusion, the value of α is 2.
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Maximum area of a rectangle which can be inscribed in a circle of given radiusRis given byαR2. Find the value ofα.Correct answer is '2'. Can you explain this answer?
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