This is incorrect. Since you have converted from a and c to functions of r, the derivatives are both dr/dt: [itex]2\pi r dr/dt= 2\pi dr/dt[/itex]DollarBill said:Homework Statement
The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference
The Attempt at a Solution
c=circumference
a=area
If the rate of change in the circumference and area are equal,
da/dt = dc/dt
πr2=2πr
2πr da/dt = 2π dc/dt
What was the question the problem asked?So would the radius just be 1?
"The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is:"HallsofIvy said:This is incorrect. Since you have converted from a and c to functions of r, the derivatives are both dr/dt: [itex]2\pi r dr/dt= 2\pi dr/dt[/itex]
What was the question the problem asked?
A related rate in a circle refers to the change in the radius of a circle over time. This change is dependent on other variables, such as the circumference or the area of the circle.
The related rate for a circle can be calculated using the chain rule from calculus. This involves taking the derivative of the circle's equation and using it to find the rate of change of the radius in relation to other variables.
The related rate of a circle is directly related to its circumference and area. This is because the radius, which is the variable being changed, is used in the formulas for both the circumference and the area of a circle.
One example could be a balloon being inflated. As the balloon expands, the radius of the circle formed by the balloon's surface increases, and the related rate of the circle can be used to calculate how quickly the volume of the balloon is changing.
One common mistake is not properly setting up the equation using the given information and the chain rule. Another mistake is not correctly isolating the rate of change variable and substituting in the given values. It is also important to pay attention to units and ensure they are consistent throughout the problem.