Suppose the circumference of a circle is increasing at a rate of 3 cm/s. Find the rate at which the area of the circle is changing when the area is 64pi square centimeters. nEEd ALot of HELP on this one. !! PLZ.
Jayden--CUHS
When the area of the circle is 64 pi cm^2, the radius is 8 cm. OK so far?
The rate of change of the circumference is 2*pi*dr/dt. Since that equals 3,
dr/dt = 3/(2 pi) cm/s
The rate of change of area is
d(pi*r2)/dt = 2*pi*r* dr/dt
= 2*pi*8*[3/(2 pi)]= 24 cm^2/s
Thanks! :)
To find the rate at which the area of the circle is changing, we need to use derivative calculus. Here's how to approach this problem step by step:
Step 1: Understand the problem
We are given that the circumference of the circle is increasing at a rate of 3 cm/s. We need to find the rate of change of the area when the area is 64π square centimeters.
Step 2: Define the variables
Let's define the variables:
- A represents the area of the circle.
- r represents the radius of the circle.
- C represents the circumference of the circle.
- t represents time.
Step 3: Identify the formulas
We know the formula for the circumference of a circle is C = 2πr.
We also know the formula for the area of a circle is A = πr^2.
Step 4: Relate the variables
The problem tells us that the circumference, C, is increasing at a rate of 3 cm/s. This can be expressed as dC/dt = 3 cm/s.
Step 5: Relate C and r
Using the formula for the circumference, we can substitute C = 2πr into the rate equation:
dC/dt = d(2πr)/dt
Step 6: Differentiate
Take the derivative (d/dt) of both sides of the equation, treating r as a function of time:
dC/dt = 2π(dr/dt)
Step 7: Solve for dr/dt
Now we can solve for dr/dt, which represents the rate at which the radius is changing:
dr/dt = (dC/dt) / (2π)
Step 8: Find r when A = 64π
Since we need to find the rate of change of the area when the area is 64π square centimeters, we need to find the corresponding radius. Using the formula A = πr^2, we have:
64π = πr^2
r^2 = 64
r = 8 cm (taking the positive value as radius cannot be negative)
Step 9: Find dA/dt
Now that we have the radius, we can find the rate at which the area is changing. Differentiate the area formula with respect to time (t) to get dA/dt:
A = πr^2
dA/dt = d(πr^2)/dt
dA/dt = 2πr(dr/dt)
Step 10: Substitute values and solve
Substitute the given values r = 8 cm and dr/dt = (dC/dt)/(2π) = 3/(2π) into the equation from Step 9:
dA/dt = 2π(8)(3/(2π))
Simplifying this expression, we get:
dA/dt = 48 cm^2/s
Therefore, when the area of the circle is 64π square centimeters, the rate of change of the area is 48 cm^2/s.