Answer:
The rate of change of the area of the circle when the radius is 4 meters = 2 meters²/minute ⇒ answer 4
Step-by-step explanation:
* Lets revise the chain rule in the derivative
- If dy/da = m and dx/da = n, and you want to find dy/dx
∴ dy/dx = dy/da ÷ dx/da = m ÷ n = m/n
* In our problem we have
- The rate of increasing of the circumference dC/dt = 0.5 meters/minute
- We need the find the rate of change of the area of the circle
when the radius is 4 meters
- The common element between the circumference and the area
of the circle is the radius of the circle
* We must to find dC/dr and dA/dr and use the chain rule to
find dA/dr
- Find the rate of change of the radius dr/dt
∵ C = 2πr
- Find the derivative of C with respect to r
∴ dC/dr = 2π ⇒ (1)
∵ dC/dt = 0.5 meters/minute ⇒ (2)
- Divide (1) by (2) to get dr/dt by using chain rule
∵ dC/dt ÷ dC/dr = 0.5 ÷ 2π
∴ dC/dt × dr/dC = 0.5 × 1/2π ⇒ cancel dC together and change
0.5 to 1/2
∴ dr/dt = 1/2 × 1/2π = 1/4π ⇒ (3)
- Find the rate of change of the area dA/dt
∵ A = πr²
- Find the derivative of A with respect to r
∴ dA/dr = 2πr
∵ r = 4
∴ dA/dr = 2π(4) = 8π ⇒ (4)
- Multiply (4) by (3) to get dA/dt by using chain rule
∵ dA/dr × dr/dt = 8π × 1/4π ⇒ divide 8 by 4 and cancel π
∴ dA/dt = 2 meters²/minute
* The rate of change of the area of the circle when the radius is
4 meters = 2 meters²/minute
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