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The area of circle is given by A=(pi)r²,where r is the radius. Calculate the rate of increase of area w.r.t radius.
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The area of circle is given by A=(pi)r²,where r is the radius. Calcula...
Rate of Increase of Area with respect to Radius
The rate of increase of area with respect to the radius is a measure of how fast the area of a circle changes when the radius is changed. To calculate this rate, we need to differentiate the area formula with respect to the radius.
Differentiating the Area Formula
To find the rate of increase of area with respect to the radius, we differentiate the area formula A=(pi)r² with respect to r. This is done using the power rule of differentiation, which states that for any constant n, the derivative of x^n with respect to x is nx^(n-1). Applying this rule to the area formula, we get:
dA/dr = d/dx [(pi)r²]
= (pi) d/dx (r²)
= (pi) * 2r
= 2(pi)r
Interpreting the Result
The derivative dA/dr gives us the rate of change of the area with respect to the radius. In this case, it tells us how much the area of a circle increases for a small increase in the radius. The value of the derivative, 2(pi)r, is a constant multiple of the radius. This means that the rate of increase of the area is directly proportional to the radius.
Example
Let's consider an example to better understand the rate of increase of area with respect to the radius. Suppose we have a circle with a radius of 5 units. The rate of increase of the area with respect to the radius can be calculated as:
dA/dr = 2(pi)(5)
= 10(pi) units squared per unit
This means that for every 1 unit increase in the radius, the area of the circle will increase by 10(pi) square units. In this case, the rate of increase of the area is approximately 31.42 square units per unit.
Conclusion
The rate of increase of the area with respect to the radius is given by the derivative of the area formula A=(pi)r² with respect to r. In this case, the derivative is 2(pi)r, indicating that the rate of increase of the area is directly proportional to the radius. This information is useful in various applications such as engineering, physics, and mathematics, where understanding how changing one variable affects another is essential.
The rate of increase of area with respect to the radius is a measure of how fast the area of a circle changes when the radius is changed. To calculate this rate, we need to differentiate the area formula with respect to the radius.
Differentiating the Area Formula
To find the rate of increase of area with respect to the radius, we differentiate the area formula A=(pi)r² with respect to r. This is done using the power rule of differentiation, which states that for any constant n, the derivative of x^n with respect to x is nx^(n-1). Applying this rule to the area formula, we get:
dA/dr = d/dx [(pi)r²]
= (pi) d/dx (r²)
= (pi) * 2r
= 2(pi)r
Interpreting the Result
The derivative dA/dr gives us the rate of change of the area with respect to the radius. In this case, it tells us how much the area of a circle increases for a small increase in the radius. The value of the derivative, 2(pi)r, is a constant multiple of the radius. This means that the rate of increase of the area is directly proportional to the radius.
Example
Let's consider an example to better understand the rate of increase of area with respect to the radius. Suppose we have a circle with a radius of 5 units. The rate of increase of the area with respect to the radius can be calculated as:
dA/dr = 2(pi)(5)
= 10(pi) units squared per unit
This means that for every 1 unit increase in the radius, the area of the circle will increase by 10(pi) square units. In this case, the rate of increase of the area is approximately 31.42 square units per unit.
Conclusion
The rate of increase of the area with respect to the radius is given by the derivative of the area formula A=(pi)r² with respect to r. In this case, the derivative is 2(pi)r, indicating that the rate of increase of the area is directly proportional to the radius. This information is useful in various applications such as engineering, physics, and mathematics, where understanding how changing one variable affects another is essential.
Community Answer
The area of circle is given by A=(pi)r²,where r is the radius. Calcula...
A=πr^2diff w.r.t.rdA/dr=π2rdA/dr=2πr
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The area of circle is given by A=(pi)r²,where r is the radius. Calculate the rate of increase of area w.r.t radius.
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The area of circle is given by A=(pi)r²,where r is the radius. Calculate the rate of increase of area w.r.t radius. for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about The area of circle is given by A=(pi)r²,where r is the radius. Calculate the rate of increase of area w.r.t radius. covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area of circle is given by A=(pi)r²,where r is the radius. Calculate the rate of increase of area w.r.t radius..
The area of circle is given by A=(pi)r²,where r is the radius. Calculate the rate of increase of area w.r.t radius. for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about The area of circle is given by A=(pi)r²,where r is the radius. Calculate the rate of increase of area w.r.t radius. covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area of circle is given by A=(pi)r²,where r is the radius. Calculate the rate of increase of area w.r.t radius..
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