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The circumference of the circle is increasing at a rate of 0.5 meters per minute. What's the rate of change of the area of the circle when the radius is 4 meters?

1: 3 meters per minute
2: 4 meters squared per minute
3: 4 meters per minute
4: 2 meters squared per minute
5: 7 meters per minute

This is actually for a game but I'm really bad at math.

Respuesta :

Ashraf82

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

abidemiokin

The rate of change of the area of the circle when the radius is 4 meters is 2m²/min

How to calculate the circumference and area of a circle

The formula for calculating the circumference of a circle is expressed as;

C = 2πr

where:

r is the radius of the circle

The rate of change of circumference is expressed as:

[tex]\frac{dC}{dt} = \frac{dC}{dr} \times \frac{dr}{dt} \\0.5=2 \pi \times \frac{dr}{dt}\\ \frac{dr}{dt} = \frac{0.5}{2\pi} \\ \frac{dr}{dt} = 0.0796 m/min[/tex]

The change in area of the circle is expressed as:

[tex]\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}\\ \frac{dA}{dt} =2 \pi r\times 0.0796\\ \frac{dA}{dt} = 2(3.14)(4)\times 0.0796\\ \frac{dA}{dt} = 2m^2/min\\[/tex]

Hence the rate of change of the area of the circle when the radius is 4 meters is 2m²/min

Learn more on rate of change here: https://brainly.com/question/8728504

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