A circle has a radius lf 10/(pi-1) which is the same value as the side of a square. Both the radius of the circle and the side of the square are gowing at 1 in/sec. Find the difference between the rates of change of their area in in/sec.
circle: a = pi r^2
da/dt = 2pi r dr/dt = 2pi * 10/(pi-1)
square: a = s^2
da/dt = 2s ds/dt = 2*10/(pi-1)
da(circle)/dt - da(square)/dt
= 20/(pi-1) (pi-1) = 20 in^2/s
To find the difference between the rates of change of their area, we need to find the rates of change of the area of the circle and the square separately.
Let's start with the circle:
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
Given that the radius of the circle is changing at a rate of 1 in/sec, we can find the rate of change of the area by taking the derivative of the area formula with respect to time:
dA/dt = d/dt (πr^2)
Using the chain rule, we have:
dA/dt = 2πr(dr/dt)
Now, let's substitute the given value for the radius of the circle, which is 10/(π - 1):
dA/dt = 2π(10/(π - 1))(dr/dt)
Next, let's find the rate of change of the area of the square:
The area of a square is given by the formula A = s^2, where A is the area and s is the length of a side.
Given that the side of the square is changing at a rate of 1 in/sec, we can find the rate of change of the area by taking the derivative of the area formula with respect to time:
dA/dt = d/dt (s^2)
Using the chain rule, we have:
dA/dt = 2s(ds/dt)
Now, let's substitute the given value for the side of the square, which is also 10/(π - 1):
dA/dt = 2(10/(π - 1))(ds/dt)
Now, we can find the difference between the rates of change of their area by subtracting the rate of change of the square's area from the rate of change of the circle's area:
Difference = dA_circle/dt - dA_square/dt
Difference = 2π(10/(π - 1))(dr/dt) - 2(10/(π - 1))(ds/dt)
Simplifying further, we have:
Difference = 2(10/(π - 1))(π(dr/dt - ds/dt))
Now you can substitute the values of dr/dt and ds/dt into the equation to find the difference between the rates of change of their areas in in/sec.