Suppose a forest fire spread in a circle with radius changing at a rate of 5 feet per minute. When the radius reaches 200 feet, at what rate is the area of the burning region increasing?
area= PI r^2
dArea/dt= PI *2r* dr/dt
r=200, dr/dt=5, compute dArea/dt
2000pi cm2per minute
To find the rate at which the area of the burning region is increasing, we can use the formula for the area of a circle:
Area = π * r^2
where r is the radius of the burning region. We need to find the rate at which the area is increasing, so we need to differentiate this equation with respect to time (t).
d(Area)/dt = d(π * r^2)/dt
Now, let's differentiate both sides of the equation:
d(Area)/dt = d(π * r^2)/dt
= 2πr * dr/dt
We are given that the radius is changing at a rate of 5 feet per minute, so dr/dt = 5. Also, when the radius reaches 200 feet, r = 200.
Substituting these values into the equation, we get:
d(Area)/dt = 2π * 200 * 5
= 2000π
Therefore, when the radius reaches 200 feet, the area of the burning region is increasing at a rate of 2000π square feet per minute.
To find the rate at which the area of the burning region is increasing, we first need to recognize the relationship between the radius and the area of a circle. The formula for the area of a circle is A = πr², where A represents the area and r represents the radius.
To find the rate at which the area is changing with respect to time, we need to differentiate the area formula implicitly with respect to time (t) using the chain rule.
Differentiating both sides of the equation A = πr² with respect to time, we get:
dA/dt = 2πr(dr/dt)
Here, dA/dt represents the rate at which the area is changing with time, and dr/dt represents the rate at which the radius is changing with time.
Given that dr/dt = 5 feet per minute and r = 200 feet, we can substitute these values into the derived equation:
dA/dt = 2π(200)(5)
= 2000π
So, the rate at which the area of the burning region is increasing when the radius reaches 200 feet is 2000π square feet per minute.
Therefore, the area of the burning region is increasing at a rate of 2000π square feet per minute.