Question

A circular cloud of poison gas from a factory explosion is expanding so that t hours after the explosion the radius of the cloud is R(t)=50+20t meters. How fast is the area of the cloud increasing 4 hours after the explosion?

Answer 1

Answer:

The area of cloud increasing 4 hours after the explosion is 16328 square meters.

Explanation:

The radius of the cloud is R(t) = 50 + 20t

We need to find how fast the area of cloud increasing when t = 4 hours.

Area of a circle A = $\pi *r^2$

Now we have to find the derivative with respect to "t"

$\frac{dA}{dt} = \pi *2r*\frac{dr}{dt}$  ----------(1)

We know that r = 50 +  20t, let's find the derivative with respect to "t"

$\frac{dr}{dt} = 20$

Now plugin $\frac{dr}{dt} = 20and r = 50 + 20t in the equation (1), we get

$ $\frac{dA}{dt} = \pi *2(50 +20t)(20)$

Now let's find $\frac{dA}{dt}  when t = 4$. Plug in t =4, we get

[/tex] $\frac{dA}{dt} = \pi *2(50 +20(4))(20)$

[/tex] $\frac{dA}{dt} = \pi *2(130)(20)$

[/tex] $\frac{dA}{dt} = \pi *5200$

We know that the value of $\pi = 3.14$

So,

[/tex] $\frac{dA}{dt} = \ 3.14 *5200$

$\frac{dA}{dt}  = 16328 m^2$

Therefore, the area of cloud increasing 4 hours after the explosion is 16328 square meters.

Answer 2

How fast  the area of the cloud is increasing 4 hours after the explosion is 16336.28 m²/s

Area of circular cloud

Since the cloud is circular, its area is A = πR(t)² where R(t) = radius of cloud = 50 + 20t

Now, the rate at which the cloud area is increasing is

dA/dt = dπR(t)²/dt

dA/dt = 2πR(t)dR(t)/dt

Rate of change of radius of circular cloud

dR(t)/dt = d(50 + 20t)/dt

dR(t)/dt = d50/dt + d20t/dt

dR(t)/dt = 20

Rate of change of area of circular cloud

Substituting the values of R(t) and dR(t)/dt into dA/dt, we have

dA/dt = 2πR(t)dR(t)/dt

dA/dt = 2π(50 + 20t) × 20

dA/dt = 40π(50 + 20t)

Rate of change of circular cloud after 4 hours

To find how fast is the area of the cloud increasing 4 hours after the explosion, we substitutr t = 4  into the equation.

So, dA/dt = 40π(50 + 20t)

dA/dt = 40π(50 + 20 × 4)

dA/dt = 40π(50 + 80)

dA/dt = 40π(130)

dA/dt = 5200π

dA/dt = 16336.28 m²/s

So, how fast  the area of the cloud is increasing 4 hours after the explosion is 16336.28 m²/s

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