Find the rate of change of the circumference of a circle with respect to the area when the area is 4π square feet.
C = 2πr
A = πr^2
C = 2π√(A/π) = 2√π √A
dC/dA = 2√π * 1/(2√A) = √(π/A)
Why did the circle go to the gym? It wanted to increase its circumference, of course! So, let's find the rate of change of the circumference with respect to the area.
To do this, we need to differentiate the formula for the circumference of a circle with respect to the area. The formula for the circumference (C) of a circle is C = 2πr, where r is the radius. The formula for the area (A) of a circle is A = πr^2.
Let's solve this in two steps.
Step 1: Express the radius (r) in terms of the area (A).
From the area formula, we can rearrange it as r = √(A/π).
Step 2: Differentiate the circumference formula with respect to the area (A).
Differentiating C = 2πr with respect to A, we get dC/dA = 2π(d(r)/dA).
Applying the chain rule, we substitute d(r)/dA with 1/(2r) since r = √(A/π).
Finally, substituting the radius value in, we have, dC/dA = 2π/(2√(A/π)).
Simplifying further, dC/dA = π/√A.
So, when the area is 4π square feet, the rate of change of the circumference with respect to the area is π/√4π, which simplifies to 1/√π feet per square foot.
To find the rate of change of the circumference of a circle with respect to the area, we can use the derivative.
Let's denote the circumference of the circle as C and the area as A.
We know that the formula for the circumference of a circle is C = 2πr, where r is the radius of the circle.
The formula for the area of a circle is A = πr^2.
We need to find dC/dA, the derivative of the circumference with respect to the area.
To find this, we'll first express C in terms of r and A using the formula for the circumference:
C = 2πr
Next, we'll express r in terms of A using the formula for the area:
A = πr^2
Solving for r, we get:
r^2 = A/π
r = √(A/π)
Substituting this value of r into the expression for C, we get:
C = 2π√(A/π)
Now, we can differentiate C with respect to A using the chain rule:
dC/dA = 2π(1/2)(A/π)^(-1/2)(1/π)
Simplifying this expression, we get:
dC/dA = π^(-3/2)√(A/π)
To find the rate of change of the circumference of a circle with respect to the area when the area is 4π square feet, we substitute A = 4π into the expression:
dC/dA = π^(-3/2)√(4π/π)
dC/dA = π^(-3/2)√4
dC/dA = 2π^(-3/2)
Therefore, the rate of change of the circumference of a circle with respect to the area, when the area is 4π square feet, is 2π^(-3/2).
To find the rate of change of the circumference of a circle with respect to the area, we need to differentiate the formula for circumference with respect to the formula for the area. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
First, let's differentiate the formula for the circumference C with respect to r:
dC/dr = 2π.
Next, let's differentiate the formula for the area A with respect to r:
dA/dr = 2πr.
Now, we can find the rate of change of the circumference with respect to the area by dividing dC/dr by dA/dr:
(dC/dr) / (dA/dr) = (2π) / (2πr).
Since we are interested in finding the rate of change of the circumference when the area is 4π square feet, we substitute A = 4π into the expression above:
(dC/dr) / (dA/dr) = (2π) / (2πr) = 1/r.
Therefore, the rate of change of the circumference of a circle with respect to the area when the area is 4π square feet is 1/r, where r is the radius of the circle.