Radius When Area of Expanding Circle Doubles

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SUMMARY

The radius of an expanding circle is directly proportional to its area, as defined by the formula A = πr². When the area of the circle doubles, the radius increases to √2 times the original radius. This relationship indicates that if the area is increasing twice as fast as the radius, the new radius will be double the original radius. Thus, the radius effectively doubles when the area doubles.

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  • Understanding of basic calculus, specifically derivatives.
  • Familiarity with the area formula of a circle (A = πr²).
  • Knowledge of proportional relationships in geometry.
  • Basic algebra skills for solving equations.
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  • Study the concept of derivatives in calculus.
  • Explore geometric properties of circles and their applications.
  • Learn about proportional relationships in mathematical contexts.
  • Practice solving equations involving area and radius of circles.
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What is the radius when the area in square units of an expanding circle is increasing twice as fast as the radius?
 
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Originally posted by tandoorichicken
What is the radius when the area in square units of an expanding circle is increasing twice as fast as the radius?
Start with the area of a circle (A = πr2) and take the derivative with respect to time. Then apply what's given.
 


The radius of a circle is directly proportional to its area, meaning that as the radius increases, the area also increases. Therefore, when the area of an expanding circle doubles, the radius will also double. This is because the area of a circle is calculated using the formula A = πr², where r is the radius. So, when the area is doubled, the equation becomes 2A = πr². Solving for r, we get r = √(2A/π). This means that the radius will be √2 times the original radius, which is equivalent to doubling the radius. Therefore, when the area of an expanding circle is increasing twice as fast as the radius, the radius will be twice the original radius.
 

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